Given that the TSA allows common lighters and matches to be taken on board, apparently not. I think aircraft cabins are designed to be "fire resistant" (for obvious reasons) though I'd be surprised if they were fireproof.

But then why don't they allow water bottles? Is a flammable liquid worse than a flammable solid like Lithium?

The entire plane interior is fire-retardant and there are lots of extinguishers around.

Given all the ways you could start a fire on a plane, it's probably better that they're engineered to survive a small fire being started anyways.