If I live in Panama, do I have to renounce or relinquish my US citizenship?

You certainly do not have to renounce your U.S. citizenship, just because you live overseas. It is not required by the government of Panama. Keep in mind that a Panamanian visa won't grant you citizenship. Should you decide to seek the Panamanian nationality, that would be another story as you would be required to sign a statement whereby your renounce the U.S. citizenship. Nevertheless, signing that document would be totally irrelevant for the U.S. Government... you would still be...

You certainly do not have to renounce your U.S. citizenship, just because you live overseas. It is not required by the government of Panama. Keep in mind that a Panamanian visa won't grant you citizenship. Should you decide to seek the Panamanian nationality, that would be another story as you would be required to sign a statement whereby your renounce the U.S. citizenship. Nevertheless, signing that document would be totally irrelevant for the U.S. Government... you would still be required to pay U.S. taxes! Some wealthy individuals opt for the somewhat inelegant solution to renounce U.S. citizenship, and avoid U.S. tax liabilities, which are incurred even if you live abroad (contrary to Panama), but that is a costly procedure: Uncle Sam won't let you give up your passport for free!