A professor I know is becoming head of department, do I send congratulations or condolences?

That would depend on how well I knew the person. If they were a very close friend of long standing, I'd probably offer condolences, just as a joke. But that would take a special relationship.

In the general case, only congratulations are in order.

If you have to ask this question of us, then don't send "Condolences".

It really is that simple.

Send a message expressing your enthusiasm for their upcoming chairpersonship, your trust that they will steer the department well, and your willingness to support them as needed to achieve their goals for the department.

At base, "condolences" and "congratulations" are saying "a bad/good thing just happened", information that the recipient was already in possession of. The important part is what you say after the condolence/congratulation.