This is a somewhat tricky question, because like many of Edgar Allen Poe's stories, The Black Cat is deeply ambiguous in its writing (and intentionally so). This is a work of first person narration, and in fairly classic Poe style, it is a deeply untrustworthy first person narrator against. The main character is deeply anguished, due to his propensity towards drinking, which leads him careening into further violence, tragedy and madness. When looking at the fire, one needs to recognize that it itself forms part of a much longer chain of events, but as to the very nature of that chain of events, there the story is not clear. Is the narrator facing something supernatural? Is this entire sequence of events retribution for the mutilation and then murder of Pluto (the first cat)? Was this entire sequence a product of his own madness and his own sense of guilt? Can we say it was a mix of both? Perhaps the fire has a fully conventional explanation, while those events involving the second cat were supernatural by comparison, or vice versa. It's tricky to say, given how Poe uses first person narration in this story.
I would suggest that there's a kind of eeriness that runs across the entire chain of events. For one thing, there's the timing: his house burns down that very night, after he'd hanged Pluto in a fit of rage. Furthermore, there's the image of the giant cat which had been burned upon the wall: a cat which was hanging. It should be noted at this point, that he does provide a rational explanation for how that image was created, but the possibility remains that the fire itself was a supernatural act of retribution. In any case, be aware that he was sleeping when the fire broke out, so he does not actually know what set it off.
Of course, the chain of events will continue after this point, with the narration finding a second cat which looks eerily like the first cat (one that, furthermore, has a patch of white fur which is shaped disturbingly like a gallows). The rest of the story follows the main character as he is tormented by the presence of this second cat, attempts to kill it, kills his wife instead, and then hides her body in the cellar wall (unknowingly entombing the cat there with her). The cat will later reveal to the police where the body was hidden.
So, what caused the fire? I don't think the story gives a single definitive explanation. Rather, I'd suggest there are several viable possibilities. As said before, the first is it was an act of supernatural retribution, on behalf of the dead cat. The second possibility is that it was the narrator himself. We know he is anguished, tormented, racked with guilt, and it's worth questioning the degree to which this entire story is ultimately a window into his own psychological deterioration. A third possibility is that it was entirely accidental. Ultimately, I think the cause of the fire is left largely unknown to us, a purposeful choice on the part of Poe.