Carpet Repair Methods
A "hole-istic" approach to rug repair
Just like that first scratch on your new Lexus, a burn or stain on an otherwise good carpet can fray the nerves! I agree... your carpet is NOT an ashtray! But, as usual, riding in on his white horse, NH will try to give you some comfort and maybe an idea or two to improve your wounded wall-to-wall!
First... call your homeowner's insurance company!
Just checking. Most insurance policies cover damage to carpets. If the burn is in an area or of an extent that repair is impractical or impossible, they may foot the bill for a new carpet!
METHODS OF REPAIR... only a little sleight of hand, maybe some GOOP, and a little luck separate you from success
The "Shaving" Method
You can sometimes improve the appearance of the carpet by using a razor blade and carefully shaving off the melted ends, taking off as little material as you can. This works best with shags and dense carpets, but only if the burn is not too deep.
The "Hair Club for Men" Method
Find an area that you can "borrow" some fibers, such as the inside of a closet, under a radiator, or, better yet, a remnant from the original installation. Using a razor, trim the damaged area down to the backing. Using a toothpick or very small flat screwdriver, apply a waterproof adhesive, such as GOOP, carefully onto the backing. Do not get any on the surrounding fibers or you'll glue them together! Press the strands of fiber into the GOOP. Let the GOOP dry for 24 hours. If necessary, carefully trim the glued-in fibers back to the level of the carpet.
If you have an aversion to using a solvent based product, you can use a clear latex adhesive caulk. I prefer the GOOP because it has more initial tackiness and is more indestructible when dry.
The "PLUG" Method
This method involves completely cutting out the damaged carpet and replacing it with a new piece. Before you start, be aware that if your carpet has faded or is very worn, the patch may be noticeable. So do the smallest patch you can to improve your chances of aesthetic success. A triangle is better than a rectangle for little patches. It's just easier to cut and easier to reproduce for your patching piece.
Carpet does have a grain. You can see it when you vacuum... ever notice how the carpet shows "tracks" when you put the vacuum in one direction, and seems to fluff up when you move in another direction? That is the grain, caused by manufacturing methods. This is more pronounced on some carpets than others, and you should observe your own carpet to see if you need to orient the patch to match the existing grain.
Many repair articles tell you to use carpet tape for this repair. Good idea if you are doing a large patch, but most patches are small, so by the time you try to force the tape through a little hole, it is a balled up useless clot of hair, dirt, and carpet fibers. If the hole is near enough to a wall so that a little rollback of the carpet is possible, use the carpet tape. If not, use good old GOOP! There is a second reason not to use the tape. Self-stick carpet tape is less moisture resistant than GOOP, so future carpet cleaning may cause a taped patch to release.
NOTE: I do not recommend rolling back carpets any more than a few feet because you may cause "ripples" in the carpet, necessitating restretching. So consider rolling back the carpet as a second, not a first, choice.
Cut your triangle around the damaged area of the carpet, and size your patch to be just slightly larger or the same size... never smaller. You want the fibers to bunch a little. Cut a small piece of burlap (or other heavy fabric) at least an inch larger than the repair area. The purpose of the fabric is to back up the adhesive so that the patch doesn't stick to the floor underneath or the carpet padding. To get the fabric in place, carefully lift the carpet up from the lining by carefully pulling surrounding fibers with a pliers. Then push it through the hole. Center it under the repair area.
Once the fabric is in place, and you have test fitted the patch and are satisfied with the appearance, start gluing by first applying GOOP to the fabric to seal it to the carpeting. Work a little under the carpet with a small screwdriver, toothpick, or other nonprofessional tool. DO NOT PUT ADHESIVE ON THE PATCH, ONLY ON THE FABRIC AND EDGES OF THE CUTOUT IN THE CARPET!! Press the patch in place, get it just where you want it, and stand a chair over the top of it to remind you not to disturb it for at least 24 hours, and not wash or fuss with it for at least 48 hours.
In closing, the only "trick" with this type of desperation repair is to not cause more damage than you already have. Remove as little material as you can and keep the glue off the face of the surrounding carpet. The Vegas odds for a visually pleasing repair decrease with the height of the pile and the age of the carpet, so you can expect to improve the look... just don't expect miracles.