No you don't have to burn them. Depends on the damage as too how much is salvageable. Salvage what you can and reuse it. ex. wood frame and replace foundation if too many webs and cocoons and if the wax moth did not cut the frames too much. Depends on the amount of damage, type of frame etc as to what is salvageable.
But again, you do not have to burn the frames and hive as in foulbrood. Just be careful of transferring wax moth eggs to another hive. A strong hive can usually handle it but a trip to the freezer will kill the eggs on the frames.