i was about to suggest the same thing. I have improved a nasty mark on the carpet with a wet razor before.
The other alternative is to cut the mark out, just a square around the mark is fine using a stanley knife. Then cut a square out of the carpet someplace else inconspicuous such as under the sofa, under a cupboard etc.....then you can glue the new square in the place of the burn. Or put carpet tap around the edges of the hole in the carpet and then press the new square onto the tape, thus securing it. The only issue then will be that the new piece looks too nice: But it'll fade in time.