Burn marks on the carpet are difficult to ignore. To remove this kind of marks from your carpet, follow the tips below:

How To Get Burn Marks Out Of Carpet

Using a small pair of scissors, cut out the burned part of the carpet.
Using a mild detergent, clean the affected area using a sponge and rinse.
Using a piece of scrap carpet cut out an area of the carpet that is larger than the burn spot. To help the new carpet blend in, be sure the color matches and the fibers run in the same direction.
Place the new piece of carpet over the burned area.
Using a utility knife, cut around the edges. You’ll also want to cut through the carpet below it. When you’re cutting through, stop before you get to the carpet padding.
Lift the burnt piece of carpet out. Check to make sure the new piece of carpet fits. If it’s slightly off, trim until it matches the affected area.
Using glue or double sided tape, apply the new piece of carpet to the affected area and let it dry for several hours. Avoid walking on this patch of carpet until it’s dry. Source: Coit

Repairing Small Cigarette Burns

Snip away the singed edges of the carpet with fine manicure scissors.
Use tweezers to pull out the burned fibers and discard the along with the singed clippings.
Use your fine manicure scissors to cut out a small amount of undamaged carpet strands from another area of the carpet that is hidden from view.
Place the clean fibers in a small dish.You will need enough new fibers to fill the burn hole.
Apply strong household glue to the damaged area where you have removed the burnt fibers.
Use tweezers to press the clean fibers into the gluey spot.
Cover the repaired area with a heavy object, such as a thick book, for several days.
Comb the repaired area with a wide-toothed comb, or plump up the new fibers with your fingers so that it blends in with the rest of the carpet. Source: WikiHow

What You Will Need

Cleaning solution of 1 part hydrogen peroxide with 10 parts water (safe on natural and synthetic fibers), use the regular kind of peroxide (the 3% kind)
Soft cloths or sponges
Water

If a scorch mark remains, soak a cloth with the cleaning solution and blot the stain working from the outside towards the middle.
Repeat as necessary until the stain is removed.
Spray with clean water to rinse.
Blot with a clean, dry cloth and allow the area to dry completely. Source: HowToCleanStuff