Is it against the US Constitution to have dual citizenship and hold public office?

It actually depends on the level. Congress, Senate, presidency, Secretary of States, and alike: yes, it is against the law.

Mayor, city council, etc: no, it's not against the law.

However, if you cannot relinquish your other citizenship (e.g. Greece does not let its citizens to apply for renunciation of Greek citizenship), it is okay.