There is some evidence to suggest that Montresor might be speaking to a priest. In the beginning, he addresses someone who, he says, "well know[s] the nature of [his] soul." A priest, certainly, would be one to whom Montresor might have made confession in the past, and, in this way, such an auditor would absolutely understand Montresor and know him quite well.
Secondly, Montresor does seem to be confessing now. In the end, he says that the events of this story took place "half of a century" ago. This means that he is now an old man. If he was in his mid-twenties when these events took place, he'd be in his mid-seventies now. It is possible, then, that he is making his final confession to a priest before receiving his last rites. If he has held onto this information for so many years, it might have been weighing heavily on his conscience, though he only now confesses (when he's so near death) because he doesn't want to suffer any consequences for his actions. After all, he said in the first paragraph of the story that, in order to exact revenge, "[he] must not only punish, but punish with impunity." This means that he felt he had to punish Fortunato and also avoid punishment himself. Now, however, as an old man possibly about to die, he may want to get it off his conscience.
Finally, the Latin phrase at the end, "In pace requiescat!" (translation: rest in peace) makes it seem all the more likely that Montresor is Catholic and that he could be speaking to a priest. It is unclear whether Montresor is speaking of Fortunato or of himself when he says this. Maybe he speaks of both! If he is speaking of himself, then perhaps he hopes he can rest in peace because it is possible that he has not rested peacefully these fifty years with the weight of this murder on his conscience. If he is speaking of Fortunato, maybe he now regrets what he did, or at least recognizes that he didn't get away with the murder without consequence. His guilt may have followed him all this time.