do you know what your carpet is made of?is it wool or olefin? if you don't know it is probably olefin.do you have any scraps of the carpet,if so pull a "zipper" off the side and use it to fill the hole left by the burn.you will need a small amount of carpet latex adhesive....good luck

I occasionally get carpet burns on my knees, but no cigarette butts, can't help you here.