Pre-2017 under the old guidance, taking action to acquire foreign citizenship or recognition of such was considered a disqualifying condition for a security clearance. However, answering that you were willing to renounce your foreign citizenship was considered a mitigating factor.
Willing is the key word here--the government wouldn't actually ask you to do this in the course of the security clearance process because renounciation of citizenship is a counter-intelligence signal to foreign states, which could lead to them targeting you for information, which the US government does not want. Further, for many countries (eg UK) renounciation is not valid unless done before certain officials in some specific process, so any forced renunciation of UK citizenship by some other country generally isn't recognized, meaning you can just reapply for another passport. Under the old guidance, you would be required to surrender any foreign passports you held as another mitigating condition.
Post-2017 new guidance proclaimed that confiscated passports should be returned immediately, and this language was added:
> By itself, the fact that a U.S. citizen is also a citizen of another country is not disqualifying without an objective showing of such conflict or attempt at concealment. The same is true for a U.S. citizen’s exercise of any right or privilege of foreign citizenship and any action to acquire or obtain recognition of a foreign citizenship.
So generally speaking, it's not a disqualifying event to hold dual citizenship, so long as the foreign country is not "at conflict" with the US and some other conditions are met.
I'm not an expert, but this is my understanding of the current status.