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Well, the MW Dictionary of English Usage is clear on this matter:

"Who also refers to words for entities that consist of people" (p. 896).

The authors of the Dictionary also mention that in the past which was also used of persons but now "is usually limited to things" [emphasis mine - Alex B.].

However, if you treat an "entity that consists of people" as one unit, then which is more common than who, cf.

The committee, who are hoping to announce important changes, ....

The committee, which is elected at the annual meeting, ....

(examples from Swan 2005).

UPDATE: Here are some examples I've been able to find (British English)

It is exactly why people have been occupying St Paul’s to protest against the behaviour of the City elite and the government who is turning a blind eye. (The Times, 2011)

The result was ignored by the Government, who locked up Ms Suu Kyi and her lieutenants for decades. (The Times, 2012)

The only people with any authority in this matter are the Scottish government who have jurisdiction on the matter ...." (The Times, 2011)

'"The Government, which had not adequately consulted on the plans, he said, was 'mistaken' if it thought the changes would be cost-neutral".' (The Times, 2012)

The Lebanese Government, which is backed by Damascus, has adopted a policy of disassociation ...." (The Times, 2012)

"Which" is the correct pronoun in American English because American speakers perceive most groups as single entities; British speakers usually perceive groups as collections of individuals and say, for example, "the team are..." and "the firm are...", but American speakers use "is" instead of "are". A "prescriptivist" will say that "who" is used only for people and not institutions, but many native English speakers don't agree with that, especially when they say things like "The company, whose main office is in New York,...". In the "The Queensland Government, who licenses several casinos,..." example, I'd say that using "who" is British English and plural, and that "licenses" should be "license".

This is interesting. I found myself fishing through an MLA handbook to get my head around this question.

From what I can gather, tchrist is right. Whose is the genitive case for the name of an agency, be it the name of a company, such as Microsoft, or an institution such as Parliament or the government. So, if I've gotten my head around this right, correct examples should include:

"Microsoft, whose new licensing scheme has ..., ..."

"Parliament, whose recent actions have resulted ..., ..."

And so, similar should work for the Government of Queensland:

"The Queensland Government, who licenses ..., ..."

As to verb agreement, the singular verb seems to agree with the pronoun in this case. Consider the following:

"The State of California, who licenses ..., ..."

"The States of California, Alabama, and Texas, who license ..., ..."

I'll enjoy any feedback that further refines this train of thought.

This is a good question. For words like "government" and "company," always use the relative pronouns which and that. As for regular pronouns, use it. This is because an organization is not a person, just another blank object in the eyes of grammar. Therefore, these sentences exercise correct usage of these pronouns.

The Queensland Government, which licenses several casinos, has decided to ignore complaints about them. One casino that is located near a bar has people throwing drinks at its employees. The employees of the casino, who have dedicated much time toward the casino, have decided to call it quits with the establishment. A company that manages casinos tried to coax its employees into staying but failed like the rest.

Grammar Girl agrees with me along with many others on this case. You can check the link out as she provides further information regarding relative pronouns.

Sources

1. http://www.quickanddirtytips.com/education/grammar/is-it-a-company-who-or-a-company-that

A government is more than the people who work in its service. It encompasses organizational units, policies, laws, and an identity that transcends the people. It's therefore reasonable to state a government is not a person, and therefore, a writer should not use "who" (which typically refers to a person as its antecedent) as a pronoun for it.

"Which" is the better choice.

If you prefer, you can scroll to the bottom and read my TL;DR. Otherwise, this shouldn't take you much more than about 15 minutes to read.

Well, first of all, I am a native speaker of English, but my visit to this thread initially was to learn, not teach. After reading some comments about Google N-grams, however, I felt that your question could be answered from an angle or two not previously given and hope to provide it with this post.

Let me first start with the basics to get everyone on the same sheet of music. You provide two examples, both of which are examples of nonessential clauses, which are set off by commas. So it would appear that this, in particular, is what is driving your curiosity — usage of "government" and "who" (or "which") when used in a nonessential relative clause. But your main question is written a bit more broadly, and thus, my answer will be a bit more broad as well. So, part of my answer may be a bit too basic for you, but I need to lay down a foundation to thoroughly answer your question. Plus, even if it might be somewhat elementary for you, it might help someone else who stumbles upon this thread. And with that, I'll begin.

Though many good sources have already been cited, I'm going to use another to help lay down some basics, which are as follows:

Rule 1. Who and sometimes that refer to people. That and which refer to groups or things.

Rule 2. That introduces what is called an essential clause. Essential clauses add information that is vital to the point of the sentence.

Rule 2b. Which introduces a nonessential clause, which adds supplementary information.

Want to know an easy way to identify a nonessential clause? Look for anything set off by commas. Anything set off by commas is likely to be nonessential.

Rule 3. If that has already appeared in a sentence, writers sometimes use which to introduce the next clause, whether it is essential or nonessential.... to avoid awkward formations.

— "Who, That, Which," The Blue Book of Grammar and Punctuation

Now let's add some details to Rule 1 using a different source:

The relative pronoun who may be used in restrictive clauses, in which case it is not preceded by a comma, or in nonrestrictive clauses, in which case a comma is required.

— "Who." The American Heritage Dictionary, 3rd ed., 1996, p. 2,038. Print.

A restrictive clause is an essential clause and a nonrestrictive clause is a nonessential clause.

And then this (which was quite surprising to me):

Some grammarians have argued that only who and not that should be used to introduce a restrictive relative clause that identifies a person. This restriction has no basis either in logic or in the usage of the best writers; it is entirely acceptable to write the man that wanted to talk to you or the man who wanted to talk to you.

— "Who." The American Heritage Dictionary, p. 2,038.

This was surprising to me because up until now, I had never read this nor had I been taught this. The league of contributors to this forum is impressive and includes university professors. My credentials don't measure up to theirs in the slightest. However, I did take four years of English in high school to include Advanced Placement English, graduated from the University of Washington, and studied journalism at the Defense Information School (DINFOS). At no point in my studies was I ever taught that "that" is acceptable usage with people. In fact DINFOS is where I was first introduced to the AP Stylebook and students of DINFOS were expected to write according to its guidelines. I don't have a recent copy of the book, but according to this source here, its guidelines still include reserving "that" for people.

Some examples of "who" and "that":

"The man who does not read has no advantage over the man who cannot read." — Mark Twain

"It is not the critic who counts; not the man who points out how the strong man stumbles, or where the doer of deeds could have done them better. The credit belongs to the man who is actually in the arena, ..." — Theodore Roosevelt

"The Man That Corrupted Hadleyburg"

— Short story by Mark Twain*

*Legendary 20th century American fiction writer, but better known for his ability to capture regional dialogue than for adhering to grammatically correct sentence structures.

Now I know that you didn't ask for general guidelines regarding "who" versus "that," but this is relevant in that it points out that differences of opinion in how words should be used exist even among reputable works of reference. In addition to that, there also exists differences between what such references prescribe and usage in actual practice, but I'll touch on that more later. Before getting to that, let's lay down a few more details about "that" and "which":

The standard rule is that that should be used only with a restrictive (or "defining") relative clause ... in this use it should never be preceded by a comma.

— "That." The American Heritage Dictionary, p. 1,859.

Some grammarians have argued that symmetry requires that which should be used only in nonrestrictive clauses, as that is to be used for restrictive clauses. Thus, they suggest that we should avoid sentences such as I need a book which will tell me all about city gardening ...

— "That." The American Heritage Dictionary, p. 1,859.

The N-grams

At the beginning of this post, I mentioned that part of what prompted me to add an answer was a discussion on N-grams, specifically the Google Ngram Viewer. One person mentioned that this was the most relevant N-gram, another said it was this one, and yet another claimed that this was the best one. I took a look at all three and then reread your question. After doing so, I believe these might be the most relevant N-grams for you:

Based on the nature of your question, I didn't really need to go back 100 years with these, but by starting the search at 1900, the charts show some patterns you wouldn't have seen had I only gone back a decade or so. The pattern for "government which" is rather interesting for starters. Even so, the most important aspect of the chart on top is that it shows how much more frequent "government that" is, compared to the other two phrases.

The chart on the bottom shows results with some type of punctuation between the two words. Based on another search I did, that piece of punctuation is a comma for the vast majority of these (98/99%?). When it isn't a comma, it is likely to be an em dash or a left parenthesis, but not a period. The use of an em dash or parenthesis serves to set off nonessential clauses in much the same way a comma would. On the other hand, a period is not likely to make much sense unless "government" ends the sentence and "who" or "which" are the start of a new one. Possible, but not likely. If you've got a computer and software powerful enough to handle huge files of data, you can take a look at the data behind these charts by visiting this page and downloading the relevant files.

In addition to the Google N-gram search, I also did some Google searches with a "News" filter. You'll see my findings from those searches in the charts below:

As you can see, the relative pronoun "which" is rarely used as the relative pronoun in a restrictive clause, but "who" can serve as either and often does. Typically, though, the relative pronoun "that" follows "government." If you saw this post when I first published it, you'll notice I made some changes to the chart above. Upon closer inspection, I noticed that a lot of results that were returned for "government who" were actually in sentences where a phrase like "a member of" (or something similar) preceded "government who," so I went back to the charts and updated them and that is what you see in the charts above now.

You didn't ask for this, but I thought it would be helpful to include. It's exactly what the chart indicates — a chart of incorrect usage. The relative pronouns "which" and "who" many times served as a nonrestrictive or nonessential clause, but weren't set off by commas. The relative pronoun "that" had the opposite problem. Here's an example of a sentence whose nonrestrictive clause was not set off with commas:

"A source in state government who requested anonymity to speak freely claimed that after Stewart-Cousins told Amazon in a meeting that she did not think Gianaris’ appointment would be approved by Cuomo, and that she planned to recommend someone else when he got turned down."

— "Why Cuomo is so furious at Long Island Senate Dems," City and State New York

This is my revised version of it:

A source in state government, who requested anonymity to speak freely, claimed that after Stewart-Cousins told Amazon in a meeting that she did not think Gianaris’ appointment would be approved by Cuomo, she planned to recommend someone else when he got turned down.

In addition to setting off a nonrestrictive clause, I had to make other changes to make the sentence readable. Ironically, the reporter who wrote this article uses this exact same phrase — who requested anonymity to speak freely — earlier in the story but does set it off with commas.

That actually isn't such a good example because the antecedent is "a source in state government," not just government. So let me show you another:

“These young people deserve the chance to present their full case against their government who is harming them and let the light of justice fall where it may,” said Julia Olson, co-counsel for the young plaintiffs.

— "Kids Climate Plaintiffs: Livable Climate Should Be Our Right," Climate Liability News

Revised:

“These young people deserve the chance to present their full case against their government, who is harming them, and let the light of justice fall where it may,” said Julia Olson, co-counsel for the young plaintiffs.

This last example isn't such a good example either because it is quoted material and, therefore, isn't representative of a news outlet's standards in writing. However, when something is blatantly wrong (or misspelled as in the case of cited printed/online material), usually a sic notation will follow. I haven't seen that used when someone has chosen to use "who" as a relative pronoun for something other than a person.

The last example on this topic I'll leave you with is interesting because I found the information in a couple of different online news articles. This element of the story was initially a quote. Here's how two different news outlets used it in the story:

"We also note that Britain's approach to the whip was recently endorsed by UK government who stated that it is 'satisfied that the rules in place are sufficient to restrict and limit the use of the whip in horse racing'."

— "Cheltenham Festival: Jockeys told whip breaches could lead to Grand National crackdown," BBC Sport

“We also note Britain’s approach to the whip was recently endorsed by the UK government, which stated that it was ‘satisfied that the rules in place are sufficient to restrict and limit the use of the whip in horse racing.’”

— "Stricter whip punishments on the cards if jockeys break rules at Cheltenham," Racing Post

Either someone gave more than one interview with slightly different wording, the notes/transcription wasn't as carefully written as it could be, or a reporter or newsroom editor decided to make some minor revisions.

By the way, I know you were interested in knowing correct American English usage and the example above is British, but when I first visited this thread I ran some Google N-gram queries for "government who," and what I discovered is that it appears that they may be even less likely to use "government who" than Americans.*

*It appears I may have been wrong about who uses "government who" more — Americans or Brits. I did a very cursory study of online news articles, and if it is any indicator of differences between American English and British English, it looks as if Americans are actually less likely to use "government who," especially American journalists and/or news editors. In speech, there appears to be little to no difference (based off of direct quotes in online news articles). Even so, whether written by a journalist or spoken by a man on the street (or, more likely, a politician) "government that" is far more frequent than "government who" (or "government which") in an essential clause. See the chart below for more:

I also did a chart for "government, which" and "government, who." You can view that chart here.

The Observations

I mentioned at the beginning that I'd share some observations with you. These observations were made while reading through the thread and they came to me as I was reading through some of the examples some contributors provided. Some discussed this notion of whether or not an entity is considered a singular entity or a large entity but made up of several individuals. I'm going to leave that topic alone because I think it's been discussed sufficiently in this thread.

Instead, the observations I'm going to share here are related to the way in which people write. The majority of the time, people write literally, especially when writing factual information. But, on occasion, they also use figurative speech or similes and metaphors or ascribe actions to inanimate objects that only humans really do. And this is where I want to borrow some examples from elsewhere in this thread. In fact, the first two Alex B. uses are perfect for this example because they show you what I mean and include a contrasting example:

The committee, who are hoping to announce important changes, ....

The committee, which is elected at the annual meeting, ....

In the first example, the relative clause is followed by "are hoping to announce important changes, ...." Who besides a human (or perhaps an animal) is capable of hoping for something? Who besides a human is capable of announcing something? Nothing I can think of, so here, "who" makes sense to me.

In the second example, the relative clause is followed by "is elected at the annual meeting." This is the use of the passive voice and in this type of construction, the actor is not identified. Consequently, the entire phrase has a far less personal quality to it and, for me at least, it makes better sense to use "which" in this example.

Alex B. goes on to show even more examples that fit well with these observations, but they're British, so let's see if I can find some that are at least half as good in American English. What about these:

"To all the girls that think you’re fat because you’re not a size zero, you’re the beautiful one, its society who’s ugly." — Marilyn Monroe

Well this first quote is not grammatically correct for a couple of different reasons, but it makes a statement, and it's a good example of how "that" doesn't always have to follow a word like "society." The relative pronoun "who" after "society" personifies it, which is perfect for the message this statement delivers.

"And since I'm the only country who seems to know how to run a meeting we'll follow my rules from here on out."— Hidekaz Himaruya, Japanese manga artist

"an organization who cares"

"Then there is a city planning department who makes parking spaces extra small and highways that encourage collisions." — Sarah Noffke (Ren: The Monster's Adventure)

In the example above, note how the author chooses to use "who" with "city planning department," but reverts back to "that" for "highways." Perhaps this author only uses "who" for people and groups of people such as those found in a department.

"We act in our own self-interest. Of course we do. Name me a person or a nation who does not." — Anthony Doerr (All the Light We Cannot See)

"Can you imagine a nation who never questions the validity of cheerleaders and pom-poms?"

— Frank Zappa

Now, granted, my examples aren’t from the likes of the highly reputable British newspaper, The Times, but please don't walk away from this thread thinking that the examples you've seen from The Times are the standard. While you may find instances in which it uses "government who," most of the time, it uses "government that" (unless it is part of a nonessential clause, in which case, it uses, the vast majority of the time, "government, which"). In fact, at least one of the examples provided by Alex B. is actually a quote — the one about the government "turning a blind eye" is from Len McCluskey, General Secretary of Unite the Union, a British and Irish trade union. For further evidence that "government who" is little used, see the chart below, in which usage between The New York Times and The Times is compared:

In nonessential clauses, "government, which" is clearly preferred over "government, who" in both American English and British English, but it appears that you may occasionally see it surface in a British newspaper (if The Times is any indicator). You can see the chart for that here.

Despite the standards adhered to by journalists and the like, unless you're writing for a newspaper or magazine, the main takeaway from this section should be this:

If you're going to go against the grain and not follow guidelines in grammar and usage books, study usage on your own until you develop your own personal preferences and understand the reason behind those preferences. Then, when you decide to break the rules, you'll know why, and be that much more confident in your word choice.

The Audience

As you may already know, the military loves acronyms and one of the acronyms we were taught at DINFOS was PAM, which stands for Purpose, Audience, Message. In this section I want to talk about "audience" for a bit. In not so many words, you conveyed in your question that you want to know which combination of words is correct for an American English audience. On the surface that's a bit of a tall order, namely because such diversity exists in such a broad category. Fortunately, deciding whether or not to use "government, which" or "government, who" isn't like deciding whether or not you need to use "soda," or "pop," or "coke". But it would help to know more specifics about what you intend to write and who you plan on writing it for.

For example, if you need to know the answer to this question because it's for a paper you need to write for a college class, stop the presses and disregard anything I tried to show you that indicates usage of "government who" (or "government, who") is acceptable. Instead, bear in mind this golden nugget of information from one of the contributors to this thread:

... 67% of the teaching assistants in American universities "would mark whose wrong in a student's paper" when it was used for things.

That was news to me, and still is. I haven't read that and I see very reputable news organizations use "whose" to refer to things other than people. In fact, I'd like to see the source material to see if it might have said "who" instead. Now that I could believe. Nevertheless, I find this statistic valuable in that it reminds us that we need to be aware of the consequences when we choose to depart from English language and usage standards. You can do so, but it isn't without risk and rarely does a student get a do-over on a paper (not at the schools I've attended anyway). With an academic paper, you've usually only got one shot, so make it count and play it safe by using "government that" (if it's a restrictive clause) and "government, which" (if it's a nonrestrictive clause).

Incidentally, this backs up what I discovered via a search of Google Scholar. Though I was not able to search specifically for "government who" versus "government, who," it was a bit irrelevant because the results so overwhelmingly favored "government that" (by more than 10:1) and none of them contained punctuation between the two words. So, I would definitely use "that" after "government" instead of "who" if I were writing a paper that was going to be graded (unless, of course, the phrase was a nonrestrictive clause, in which case I'd use "government, which" instead).

But if your audience is not a professor, and instead it's a news editor or publisher, take the time to find out what stylebook it adheres to or what publishing guidelines and formats are required. Has the agency developed its own in-house stylebook or does it use the AP Stylebook? While the AP Stylebook is something of a bible for a lot of news outlets, many of them have also developed their own rules about style and if you want to impress them, it would behoove you to get a copy of it and make sure your writing doesn't violate some basic rules that have been laid out in it. If you can't manage to get a copy of one or you have a question that it doesn't seem to answer, go visit the agency's website and see for yourself what their style is like. Does it tend to hyphenate words or not? Which ones? What about capitalization? Does it use the "Oxford comma?" And nowadays there's the issue of gender and which pronouns are acceptable to use. Has the agency you plan to write for developed a policy on it?

If you're not writing for a professor or a news editor, then what you want to do is just think about your audience and who it is. Are these people you know or strangers you may never meet? Answering that will go a long way in determining how formal your writing needs to be. If your writing doesn't need to be that formal you can get away with unorthodox wording and bucking some conventional rules as long as your message gets conveyed. Does your audience like rules and tradition? Or does it prefer creativity and breaking the rules?

Another thing you may want to ask yourself is What types of publications does my audience read? Do they read the New York Times or BuzzFeed? If the former, I'd go with "government that" and if the latter, you can probably get away with "government who" or "government, who" as it did in this article here. But don't think a lowbrow newspaper isn't without its standards. Take the National Enquirer, for example. Though many may question its choice of content and reporting methods, this is definitely a paper that adheres to standards in terms of journalistic style. You'd have difficulty finding "government who" in a National Enquirer; it uses "government that."

Why the debate?

Until I took the time to try to answer this question, my notions regarding whether or not one could use "who" as a relative pronoun for collective nouns representing groups of people (e.g., government, society, department, ...) was rather fuzzy. Clearly, though, I had left open the possibility in my mind that there might be instances in which "government who" (or "government, who") might be acceptable. But after doing research to try to help answer this question, I have to wonder why this is the case because clearly the uncertainty I had about usage on this matter is shared by others based off of the fact that you've asked the question, which has been viewed, to date, 13,467 times and the fact that there is clear evidence that some use "government who" or "government, who," especially in British English.

I suppose the best way I can answer that is to say that beyond the world of academia and journalism, most of us are never required to write according to a standard. Therefore, we are free to pick and choose the authoritative sources at hand that determine what constitutes correct usage and what does not. But the vast majority of us likely don't even do that. Instead, writing is largely a product of what we are taught in school and what we hear and read. Information received through education or our own personal listening or reading choices can vary widely. In just this thread alone, I see at least four answers that advise against using "who" with a word like "government," but one (and the top answer at that!), not only claims that it is acceptable, but cites a reputable source. In addition to that, you'll find things like this online:

BBC Learning English

which is yet another reputable source claiming that one can consider a collective noun representing a group of people as a single entity or a group of people and, therefore, both "which" and "who" can be considered acceptable.

I'm not going to belabor the point by continuing to cite sources that do find "government, who" to be acceptable and those that don't. But, in general, and despite the cited material from Merriam-Webster that Alex B. provides, I think this passage sums it up best and echoes something one of the contributors to this thread has written, which is:

In BE, ... government [is considered] singular or plural ... Speaking of government as a set of individuals, we use who; speaking of government as an impersonal entity, we use which. In AE ... government is singular and therefore requires which, not who.

— panjandrum, occasional moderator for WordReference, in a discussion thread titled,

"The government: who or which?"

TL;DR

"Who" and sometimes "that" refer to people. "That" and "which" refer to groups or things.

The relative pronoun "that" is typically only used to introduce a restrictive clause — a clause essential to the sentence and, therefore, not set off by commas.

Another word for restrictive clause is "essential clause"; another word for nonrestrictive clause is nonessential clause.

A restrictive (or essential) clause is not set off by commas; a nonrestrictive (or nonessential) clause requires commas.

Of these combinations: "government that," "government which," and "government who," "government that" is the most common pairing by far.

Of these combinations: "government" + "which" and "government" + "who," "government" + "which" is the more common pairing, but only in a nonrestrictive clause and will, therefore, need to be set off by commas.

The relative pronoun "who" does follow "government" as part of either an "essential clause" or "nonessential clause" in actual usage, but many grammar and usage books, as well as the AP Stylebook, inform readers that "who" should be reserved for people and "that" and "which," for things and objects.

Not setting off a nonessential clause with commas can make a long sentence very difficult and cumbersome to read.

If you are going to use "who" with an inanimate object or entity of some abstraction (such as government), have a reason for doing so. Either you see the entity as more of a group made up of several individuals rather than a single entity, or the relative clause referencing it imparts some sort of personification to it.

If you're writing an academic paper (or a news article), use "government that" (in a restrictive/essential clause) or "government, which" (in a nonrestrictive/nonessential clause).

Okay, okay, native speaker of American English to the rescue, if belatedly!

The answer is "which", never "whose". Americans see governing bodies or sports teams as things, not as a person or persons. So, two example sentences:

The US Congress has held a meeting this morning to discuss Damascus.

The whole Boston Red Sox team is hosting a free beer night.

There. The final word.