Burn marks can occur on a carpet if someone dropped a cigarette or match on it. It can also get burnt if any candle wax drips off of the candle on to the floor. You may think that this is impossible to get up, but it isn't.
Use a razor to remove any burnt tips in your carpet fibres. This works for plain burns. Just lightly work the razor out and up on all sides of the carpet fibres. This takes patience, especially if you burnt a large area of your rug. It does work though.
Cut a brown paper bag to fit over your wax burn. Turn your iron on low and iron over the brown paper bag. Keep checking because you may need to replace the bag a few times. The wax melts right off of the carpet and attaches to the paper. You can then freshen up your carpet fibres with a razor just like you did in step 1.
Use the blade of one side of a pair of your scissor and run it across the burn. Go back and forth several times until the burnt parts of the carpet begin to flake off. It will begin to look much better than it did before because the healthy fibres underneath the burn will begin to show.
Cut the burnt fibres off with a pair of scissor. Then cut some good fibres from some leftover carpet or carpet in your closet that no one will see and superglue it in the spot where the burn was. The superglue is powerful and will keep your new carpet fibres in place even when you vacuum over them.
Use a piece of sandpaper to sand away the burnt part of the fibres. You can then determine whether the fibres look normal again or whether they are burnt beyond repair. If so you can cut them out and replace them with new carpet of the same colour and texture.
You can also freeze wax burns with a few ice cubes and then chip them away with a spoon. Carpet glue and a hot glue gun also work to secure new fibres in the old fibres place.
Don't put your iron on high or you could set your brown paper bag on fire.