Our only clue to whom Montresor is speaking is found in the second line of the story, when Montresor says, "You, who so well know the nature of my soul, will not suppose, however, that gave utterance to a threat." It would be fair to assume Montresor is making a confession to another person who knows him well, perhaps a priest. The story that Montresor tells, his murder of Fortunato, takes place during Carnival, the night before the first day of Lent. Because he calls his family home a palazzo, it is fair to assume it is in Italy, where Carnival originated. As a result, it is plausible that Montresor is a Catholic confessing to a priest.
Furthermore, the story's final words, "In pace requiescat!" are Latin for "rest in peace!" Latin is the official language of the Catholic Church.
It should be noted, though, that Poe was not a Catholic; not much is known about his personal religious faith or lack thereof.