Although Poe does not ever actually state what the narrator is guilty of, we can make inferences based on some context clues:
- First is the (Spanish) Inquisition in and of itself: it was established in order to keep up the proper method of practicing Catholicism within the kingdoms of Ferdinand II of Aragon and Isabella I of Castile. So from that, we can infer that the narrator's crime was something that went against the Catholic church.
- Next is a certain term the narrator mentions at the beginning: autos-da-fe. There were multiple kinds of that specific ritual, often in the form of public penances, but it was not unheard of to have a private one, which is surely what our narrator is facing. These did not always end in death - in fact, they usually just ended in the person publicly returning to Catholicism, but punishing the unrepentant heretics happened. (Of course, it is highly unlikely that such a horrible torture method as the narrator faces was ever used, but not much about Poe's story is actually historically accurate.) Generally, autos-da-fe were used to punish heretics and apostates, so people who either said blasphemous things against the beliefs of the church or people who renounced the church entirely. So we can assume that the narrator's crime was one of those two things.
In the end, we cannot say with certainty what the narrator's crime was, but we can be sure that it had something to do with going against Catholicism in some form or other, which was the main focus of the Inquisition.