Unfortunately, there is no absolute answer to this question, as Poe never specifically states what the narrator is guilty of. However, Poe does leave enough clues for us to make an educated guess: he he gives us the location and the circumstance, thereby telling us the time period, and he tells us the method.
On multiple occasions, the narrator mentions the Inquisition. Now, there were multiple Inquisitions at one point or another; there was also a Portuguese Inquisition and a Roman Inquisition, but by mentioning Toledo, we can know that it is the Spanish. At the beginning, the narrator mentions something called an auto-da-fe, which he says is how those condemned to die were executed. Toledo did have quite a few auto-da-fe, so that confirms things more. And by telling us at the end that General Lasalle saved the narrator, Poe tells us that it is sometime at the beginning of the 19th century.
As for what the narrator was guilty of, the auto-da-fe was usually the penance for condemned heretics and apostates (someone who spoke against the church's beliefs and someone who separated themselves from the church respectively), so we can be fairly certain that the narrator was guilty of one of those things.