Do interviews require taking vacation days? When interviewing for a new position, is it common practice to count the days spent on interviews as vacation days?

It wouldn't seem unreasonable to count these as vacation days, except that I could also see them as being work days. If I were only visiting to give a lecture and speak with faculty in another department, it would seem like work, and presenting at a conference is certainly not vacation.

A few follow up questions - does it matter that I am a post-doc (e.g. would it be different from a faculty, in which case interviewing for another position would not be in the interest of the current employer)? Would it be different if I were interviewing at a company rather than a university? Does it depend on the country? I am in the U.S.

(this is related to the question related to Do presentations given during interviews count as invited talks?)
You should ask your advisor, but it is reasonable for an advisor to allow you time (potentially several weeks or months) to essentially be away from the lab and your research in order to interview for jobs. In my opinion, this is a concession that is reasonable to give as a reward for hard work during previous phases of an advisee's research. If not, the advisor likely does not have the advisee's interests in mind. That could come back to bite him/her later when attempting to find new advisees.

On the other hand, if you're spending a lot of extra leisure time on the trips that is not part of the interview, then it should probably be considered vacation time.