It's actually pretty easy to hide cigarette burn in carpets, especially short pile. And you can do it for pennies, too.
The secret? Dollar store nail polish.
It's best to start with a clean, dry carpet. Aside from that, no other prep is necessary. Most dollar stores have a wide variety of nail polish in every color you can imagine. For the burn above, there wasn't a "beige" color available, so I had to figure out how to mix a couple of colors together to get a light brown beige.
Orange and green equals brown, so a light flesh color and a dull brownish green were my choices.
Step 1: Dollar Store Nail Polish Works Miracles
The burn still leaves a great deal of the nap of the carpet behind, and the nail polish sinks into the nap. Use repeated coats, and feel free to mix colors together to get as close as you can to the tone of the carpet.
If you are using more than one color of nail polish to match the rug color, you can actually pour a drop of each color into the burn mark and mix it right there. Just don't slosh too much of it into the surrounding carpet.
This picture above was a dull greenish brown nail polish with a flesh color nail polish mixed in to match the tan color of the carpet. There's no Photoshop trickery involved.
Step 2: Use Dull Nail Polish Colors
The nail polish will soak into the burn mark, so a second coat is usually required.
You don't have to be an artist, because most carpets are actually made up of several similarly-toned colors. As a general rule, nobody is going to examine your carpets very closely. Burn marks tend to really stand out, however, and this technique makes them MUCH less noticeable.
And it's extremely durable, stands up to vacuuming and carpet cleaning without any problems, and this fix costs pennies.
Step 3: Extraordinarily Durable and Will Last As Long As the Carpet Itself.
"Carpet burns? WHAT carpet burns?"
I really didn't sweat the effects in the photo directly above too much. This is in the smoking lounge of a nursing home. With repeated applications of nail polish you can really fool the eye completely.