How can I mount a shelf above a closet door?
I did this once when I lived in an apartment and could not modify anything.

Since I had metal working and welding experience, I created a steel frame using 1 inch angle steel. Basically, it was a 'table' that stood inside the closet and it's dimensions where precisely the inside dimensions of the closet; ie; the legs were flush against the walls of the closet.

The 1 inch steel angle iron was small enough not to interfere with the closet itself and were strong enough to support the shelf (two pieces of 1/2 inch thick plywood; each 1/2 the width of the closet so they can the inserted).

I made the steel frame so that it could be assembled using screws once its put inside the closet.

When I moved out of that apartment, I easily removed the shelf assembly and viola! I had a un-modified closet to return to the landlord.