Here are some suggestions:
a. Ask the shop where you bought the carpet.
b.Do you have any offcuts of this carpet? if so you could use a very sharp Stanley knife to cut out the burn, use it as a template to cut a matching piece from the off cut and then glue the new piece into the hole. You need really strong adhesive and a second person to hold the gap open so that fibres from the original carpet do not stick to the underside of the patch. I have done this in my daughter's room with good results.
c.On a long pile carpet it might be possible to just cut off the ends of the burnt fibres
Failing that leave the rug where it is!
Good luck.