What should be done about a temporary relocation of an amenity for rebuilding?

The golden rule is to tag what's currently on the ground. So retag the temporary location as amenity=school and the site under redevelopment as a construction site (perhaps: see below!). Name both the same while the work is being performed if they have the same name on the ground. It might be wise to record a bug with OpenStreetBugs reminding yourself or others to undo this change later on, when work is complete.

If the site under redevelopment is currently just school offices with work being carried out on them while school staff go about their everyday office work, but classes have been moved for safety reasons, tag the old site as landuse=commercial (i.e. offices) and still name both sites the same.

Couple of other observations:

• This sort of situation and others like it is a suitable use for the description=* tag, which can be used for additional information which may be visible when the map is rendered. Keep such notes brief and descriptive of the current situation. Other mappers will appreciate FIXME=* or note=* tags, which are used for internal notes and can be a bit more long-winded. The name tag is for names only, not descriptions.

• Temporary works are always tricky to represent - how long must a temporary situation be in effect before you decide to alter the map? - but as noted in Jochen's excellent answer here, it's certainly OK to do this when the "temporary" alteration will be actually a feature in the landscape for several months. A school year is such a period.