When you are faced with difficult carpet cleaning problems like burns, stains or any form of discoloration, you should be aware that by following some basic carpet cleaning steps, you actually will be able to restore your carpets to their original glory. If you have a problem whereby you have tried using a spot remover on your carpet, only to find that the original color of your carpet has been altered, then you could try using an acrylic paint or felt tip marker to slowly blend the color into the fibers.
Sometimes you will find that there are deep dents in your carpet that have been caused by furniture sitting in a certain position for too long. These can be obliterated quite simply through the use of a steam iron which should be held close to the carpet, but not on the carpet (as that could damage the carpet's fibers, thus melting them). You can scrape the fibers with the edge of a coin and snip any loose threads to make it level but don't pull the thread out as this could lead to an unwanted carpet unraveling.
For burning, you should cut out the area and substitute it with a patch you have remaining from excess carpet, securing it with double-sided carpet tape or latex adhesive, unless there is a small area that has been burned. In this case you can chop off burned fibers, and glue the opening, snipping fibers from a scrap of carpet or a piece of carpet you don't see (underneath a piece of furniture for example). Once the glue is tacky you can poke the fibers into place but if the burn hasn't gone all the way through to the back, snip the burnt tips of the fibers.