In what situations would "self-plagiarism" be authorized by a professor? A lot of universities say "Cannot submit this work in whole or in part to another class without prior written consent by the instructor." 

What is an example of a situation where a student would get consent for this?
I first note that the idea that students' work is somehow not a part of their personal body of work, thus owned by them, and re-purpose-able however they like, has always struck me as very strange. In fact, this seems to me quite contrary to any notion that universities are creating intellectuals and scholars. That is, there is a powerful implicit assumption that students would never do anything at all if not commanded to do so by instructors in officially scheduled courses, so somehow do not deserve credit for any work, and do not truly "own" it.

I myself have absolutely no objection to anyone's re-use of work they happened to do before I asked them to do it. :)

All the better, in fact, if they had the sense and drive to do it before being asked.

I much prefer the idea that each student develops a body of expertise, often expressed in write-ups of various technical sorts, if only as notes-to-self, which can be brought to bear in various situations. Not that everyone is required to give up all fruits of their own prior work, to avoid some highly commodified, artifactual stigmas about "self-plagiarization". Sometimes the answer to a question does not change just because it is asked repeatedly.