One thing I love in the creature-design tag is that very often you ask for something that seems fantastic, but in fact that already exists.
That creature is us. We can put fire in our mouths (don't you do it at home though!) because our wet mouths provide some short term protection. There is a circus trick that is based on that.
But seriously though, your mouth can handle hot things for longer than some other sensitive body parts such as your fingertips.
On a more serious note: we wouldn't be humans if we had not tamed fire. Cooking our food breaks down many proteins and some fats and fibers, making them easier to digest. This allows us to meet our dietary needs with smaller intestines and other digesting organs, which frees up more resources for a larger brain. Just compare the anatomy of a chimp and a human - chimps use much more energy for digestion. They have much larger intestines too.
I would daresay that when hominids first walked the Earth, the biggest difference between them and other animals was not the enlarged brain - elephants and whales have that too. The real difference is that we don't eat raw food (except for gastronomical reasons!).
If you don't like my answer, and instead still wish for a non-human creature that does ingest fire or fuels, consider what I said about breaking down proteins and fats. Your creature will spend less of its own energy for digestion, which in turns allows it to grow tougher or more intelligent than its closest related species which do not eat fire nor fuels.
For clarification, I am suggesting eating fire - or fuels to get that fire going inside the stomach - alongside regular food.
Also consider that coal has a lot of energy in it. It is used to power cities to this day after all. It's just that no lifeform has ever evolved to extract energy from it through physiology. If any lifeform could do that, they would be to us what we are to anaerobic bacteria.