If you're now living in a new country and have a US passport, and decide to take on the citizenship of said new country - sometimes you're unable to have dual citizenship. In this case, let's say you've giving up your US citizenship; what must you do with your US passports? Do they remain active until they run out?
In practice, no matter the specifics, if you manage to hold onto a seemingly valid passport, you should be able to use it to travel, certainly abroad. But if you explicitly renounce your citizenship, you might be asked to surrender your passport or it might be added to some list of revoked passports (there are also a few international databases of stolen documents, e.g. through Interpol or the Schengen Information System but I don't know how widely used they are or if revoked passports – as opposed to stolen or lost documents – are routinely added to them).
I know people who have a Japanese passport and are still treated as Japanese citizens by the Japanese government even though they acquired another nationality. My understanding is that formally their Japanese nationality has been automatically revoked but their government simply does not know it. The authorities apparently aren't very aggressive about enforcing this rule (e.g. one of my friends travelled to and from Japan and renewed his passport from an embassy, all without any of the visa stamps/stickers you would expect if he had no other passport).
Conversely, I have heard about people whose naturalization was cancelled because they retained or regained their previous citizenship (it was in Germany, a few years back). Before the relevant ministry initiated inquiries, those people would have held both passports (or ID cards) and would have been able to vote, etc. as if they were citizens of both countries (which, as far as German law is concerned, they actually weren't).
In any case, you either are or aren't a citizen under the law but as long as the authorities don't notice it you might be able to cheat and enjoy one citizenship a little longer. Save for the databases mentioned earlier, there is just no easy way for a third party to check the validity of a passport you haven't been forced to surrender.
Now, in the specific case of the US, simply gaining another nationality is not enough to cease being a US citizen. You need to explicitly renounce your US citizenship. There is a whole procedure including two interviews with a consular officer and it seems you have to present your most recent US passport, which will be “canceled or annotated to reflect loss of U.S. nationality”.
I believe that you are required to surrender all your active passports as a part of renouncing citizenship. If you don't surrender it by claiming it was lost or stolen, then the passport is probably marked as lost or stolen, in which case you may have trouble using it in other places.
When United States citizens renounce their citizenship, the consular officer will punch holes in your passport just like when they void an old passport when giving you a new one.
There was an article in the Korea Herald about this several years ago.
Herald: In 1997, you gave up your U.S. citizenship and became a full-fledged Korean. What made you decide to do that?
Robert Holley: That's a long story. Actually, the first time I saw Lee Han-woo on TV, my wife told me he had Korean citizenship. [Imitates wife] "Isn't that wonderful!" I remember thinking, "What kind of crazy guy would give up his citizenship to stay in Korea?!" So I got into broadcasting and I was running into all sorts of problems working and I had been in Korea for the longest time. My wife said one day, "Why don't you think about changing your citizenship?" So I started to think about it seriously and I sat down with these others (Lee, Daussy, et al.) and said, "How do you feel about this?" They said, "Look. This is where we live. This is where we work. Don't you want to enjoy the full rights of all the Koreans you live with? This is where you are. If you didn't want to be here, you wouldn't be." My biggest problem with changing my citizenship was that I love America. I love being American. I also heard this horror story about a woman who gave up her citizenship. She said you have to sign this vow and they take your passport away and they punch holes in it. They say, "You shouldn't do this. You should reconsider this. You may never go to the States again. We don't have to give you a visa, you know." It scared me to death. Eventually I found out it was possible to get a U.S. visa after giving up your citizenship. So I did it and I've never had any problems.
(Emphasis mine, story archived here.)
You get to keep the canceled passport. This is important because it contains visas that were issued to you that may still be important -- even if not valid. The People's Republic of China requests all of the previous visas they have issued you when granting you another visa.
Generally the passport is considered void. The US does report cancelled/voided passports to International databases - that's how they attempted to trap Snowden in Hong Kong (he managed to pass HK border control minutes/hours before the State Department reported that his passport was revoked). Most, if not all, countries check the validity of the passport during border crossing.
If you get stuck in a foreign country using such a revoked passport you may get into trouble and your "new" country may not be able to help you (since you've effectivelly illegally entered the country you got caught at and didn't use your new passport. Some countries only allow consulates to represent their citizens only if they used that countries' passports to enter).