I've had to deal with this a couple times with my dog (sometimes he gets an upset stomach, and can't go eat grass, so he starts eating whatever he can find... including carpet threads...), and I've managed to deal with it without the landlord both times by using a carpet repairman and patching in some carpet. This will depend on how big an area is affected, but you can usually make it work.
The trick is, you have to have matching carpet to patch in or it will look wrong. The best way to do this (assuming you don't have a swatch of extra carpet laying around) is to find some carpet that looks "pretty close" (I recommend a carpet remnant store) and have the carpet repair guy cut out the carpet from the closet, replace it with the "pretty close" carpet, and then patch in the closet carpet to replace the damaged carpet. The idea being that a seam between two different colors of carpet looks natural in a closet, but not so much in the middle of the room.
You can usually tell the difference immediately after it's been patched, but after a day or so you can vacuum/clean it, and it usually evens out a bit... the longer you do this before you move out, the more chance it will have to "wear into place".