Depends on what country. If you're a dual citizen with China/Russia/hostile power, you'll probably be forced to renounce or be denied a clearance. Folks from allied countries (NATO) often don't have to renounce it; I believe they ask if you'd be willing to, but you don't have to do it unless they ask

Not true, you can hold a clearance and stay a dual citizen.

It depends...some organizations require dual citizens to renounce their foreign citizenship before they get their clearance.

Not sure why the down vote. It does depend on the sponsor, which country you hold a dual citizenship with, and what level of clearance you are trying to obtain.

The below poster is incorrect. In order to move forward in the process in some cases you will need to formally renounce and physically destroy you foreign passport in front of your investigator.

I know that sounds bad, but keep in mind no one is forcing people to obtain a clearance.

From what I understand, the worst case is that they confiscate your other passport(s) to ensure that you do not enter or exit the US with a non-US passport. I could be wrong though.