I remember back in my dorm room days. I put a poster up with a bunch of that double sided foam tape on some drywall. The day I took it down, I ripped the HELL out the drywall.
I wasn't about to pay a few hundred dollars so they could replace the drywall, so I went out, bought a box of plaster of paris, and smoothed the whole thing out, then paid one of the art guys down the hall 5 bucks to match the color and paint it. Hey, for about 15 bucks total, I saved myself a few hundred the housing people would have charged to do the same thing.
Anyway, as far as the door goes, I would try to do something with it first. I mean, if you're going to have to replace it anyway, it's worth a shot.
If the break was pretty clean, you could try and get the piece or two out of the door, buy a small block of wood to put behind the break, and glue the pieces to it, then put a little bit of wood filler around the edges and a little paint work... they might not notice. If the door matches a closet door, try setting it up so the break is on the inside of the closet, just switch the doors if the handles and stuff line up. If it's on the inside of a closet, people checking out the place when you move out will be more likely to just look to make sure the inside of the closet is clean, and might not notice a touch up job on the door.
If the door is beyond repair, then you're better off just getting the maintenance man to order one from a local store, and seeing if you can go pick it up/install it. If you don't and just let them charge you, they will overcharge for the door, charge you labor fees, and probably some other small crap penalty.
Good luck buh-day