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Cut it down to the adhesive base of the carpet and lift it out. Try to cut in a square or rectangular shape that will be easy to replicate. Using the burnt section of carpet you just cut out as a template, cut out a swatch of carpet from a less-noticeable area, like the back of your closet. If your carpet has a pattern, make sure it exactly matches that of the burnt section. You could also snip a section from loose carpet samples, which you might have left over from when the carpet was put in. Cut out a swatch of woven backing that’s about the size of your new carpet swatch. Spread strong, permanent glue on the back of the piece of backing and the new carpet patch, and along the edges of the area where the burn was. Allow the glue to settle in until it’s slightly tacky. You can get woven carpet backing online and in most home improvement stores. Slide the backing into the area where the burn was. You may be able to lift up the surrounding carpet to fit it more securely. Then, set your new carpet patch on top of the backing backing and gently press it into place. Use a small, sharp pair of scissors to cut loose any fibers on the ends that look out of place. With a small, fine-toothed comb, gently brush the new fibers to match the surrounding area. Use a large book or heavy pot to hold the new swatch down. Allow it to dry for the rest of the day and night.

Summary:
Cut out the burned area with a sharp razor. Snip out a section of carpet from an inconspicuous area. Spread glue on a piece of carpet backing and the new carpet patch. Place the backing and the new carpet patch into position. Snip off loose fibers and comb the patch to blend it in. Place a heavy object on the area and let dry for 24 hours.