Call professional help as burn marks are difficult to remove by home remedies.

Cut the surface fur of the affected area with a small pair of scissors and clean the area with detergent and water. The lower part of the burnt carpet would not be visible. Dry it with a hair dryer and brush it delicately. I am sure you will succeed in getting the look back!

Try cutting the burnt threads carefully and cover it with a piece of furniture or a small rug. You can also cut off the area completely and get it patched with a similar looking piece with the help of professionals.