There's no clear indication in the text about who Montresor is telling the story to. It leaves it up to the reader's imagination.
Montresor is telling this story to an unnamed person. We don't get a clear identification of who this person is in the story. It could be someone he trusts, or perhaps he's just unburdening his conscience to a random listener.
Maybe he wants to get something off his chest or seek some kind of validation.
The recipient of Montresor's tale is ambiguous. It might be someone he considers a confidant, or it could be a symbolic act of sharing his burden with an imagined audience to unburden his conscience.
One possibility is that he could be telling his story to a priest. In the context of seeking absolution, Montresor might be confessing his dark deed. Priests are often the recipients of confessions, and Montresor, burdened by his conscience perhaps, could be unburdening himself to a man of God.
It could be to a close friend or confidant, someone he trusts to listen and understand.
Well, it could be someone close to him, like a confidant or a friend.
He could be confessing to a close friend. Someone who he believes will understand his motives, or at least keep his secret. A friend who has known him for a long time might be more likely to accept his confession without immediately judging him harshly.
Montresor is vengeful and calculating. He plans his revenge on Fortunato meticulously.
Montresor might be sick due to the inner conflict and remorse. His conscience could have finally caught up with him, causing physical symptoms. Another possibility is that the intensity of the emotions during the events at the end of the story overwhelmed his system.
He could be confessing to a priest. In many literary works, a character might confess their sins to a religious figure. Montresor's act was a heinous crime, and confessing to a priest would be a traditional way to seek absolution, though there's no clear indication in the story that it's a priest.