What costs should be covered when inviting a university professor from another country to attend your university? What costs should be covered when inviting a university professor from another country to visit your university?

For example, imagine you wanted to invite a university professor from Harvard to a European university.
I don't think this question has any absolute answer. This depends on the professor (a "famous" speaker for a large public event will be a different issue than a "regular" professor that comes to give a talk for the students and faculty of the university), the host institution (a professor might be happy to go and give a talk at ETH Zurich, and maybe less so to travel to an unknown university), how long the trip is, and whether there are some pre-existing relations (i.e., whether the inviting professor is already friends with or at least an academic acquaintance of the speaker).

Minimally, it is standard that all costs of the speaker are covered, including (sometimes) business class airfare and a nice hotel for the duration of the trip. If the invited talk happens as part of a conference, the conference fee or entry should be waived for the speaker. Further, it is customary at least in my field to give the speaker a small (local) present, such as a bottle of good regional wine, as a sign of appreciation directly after the talk.

Whether the speaker asks for money on top of this depends on all the things above, but this seems to be relatively unusual in my field. The only persons that actually get paid in money to come and give a talk are well-known public figures, like Tim Berners-Lee. I have recently had the honour to listen to a talk of Don Knuth, and rumor has it that not even Knuth was asking for direct payment.