Can professors have romantic relationships with post-docs?
Policy on this may vary from institution to institution, so those would need to be specifically consulted.

However, the whole "when is a romantic relationship taboo?" question hinges most importantly on one thing: does one of the individuals involved exercise some sort of power over the other. It would be odd for a professor to have a relationship with one of his or her own postdocs, where the postdoc will likely need a letter of recommendation or decision on financial promotion from their now-conflicted lover. I think, generally, there is usually not a policy against this but might be frowned upon. 

This is why it isn't a good idea for a president to sleep with a white house intern, or a teacher to have a relationship with one of their current students. Even if it were technically legal, and no bias was ever realized, the potential for bias and conflict is too strong.

A professor in one department and a postdoc in another department should present no problem though, if there is no potential employment/financial promotion conflict, e.g. English Dept. vs Physics Dept.

Another question to consider is whether the relationship precedes both individuals being a member of the institution. For example one member of a married couple gets a job at the institution and the other enrolls in school there. But even in this situation, I can see potential for conflict if they are in the same (or closely related) departments.

As long as everyone is open, honest, consenting adults, and clearly discusses potential conflicts with the appropriate personnel then there may be no issue though. Trying to keep it secret, especially if there is potential for a conflict of interest is probably not a good idea.