In the linked question, we decided to deprecate the students tag, since over half the questions on this site involve students in some capacity.
The professors tag has similar issues; it is so broad as to be meaningless. Questions tagged professors range from questions about teaching by professors, to research done by professors, to becoming a professor, to communicating with professors, to misconduct by professors, and much more.
So, what should we do? I'll make my proposal as an answer, but other suggestions are welcome.
Whatever we decide, there are so many questions with this tag that we probably won't retroactively change all the historical questions, as we are doing with students. Rather, we can define what the tag should mean going forward, and we can fix non-compliant historical questions as we come across them.