The conflict in "The Cask of Amontillado" is between Montresor, the protagonist, and Fortunato, the man who he seeks vengeance on. Montresor will not allow Fortunato to continue living "with impunity" after the "insult" he dished out against the protagonist. This battle is man versus man, and that is the sole conflict in the story.
In the exposition, the conflict is set up at a carnival, representing chaos and illusion. It is not even clear if Montresor has found the right victim since he is costumed and drunk, but he lures Fortunato with compliments the reader knows are insincere.
In the rising action, Montresor leads Fortunato to his home and to the underground vaults. The conflict here is whether Fortunato will follow, and whether Montresor will continue with his evil plan.
The climax of the story is when Montressor finally admits to Fortunato, as he shackles him in the dark tomb of his catacombs, that he hates him and seeks harm against him. As Montresor slowly bricks up the chamber, Fortunato pleads for his life and the conflict between the two men has not been resolved. Will Montresor have a change of heart and realize his ghastly plan has already wrought revenge?
The falling action of the story occurs when Fortunato's screaming finally abates. As Montresor walks off, throwing his torch into the catacombs, he hears the last scream.
The story is resolved when Montresor confesses his crime to someone, although it is not clear who, years later. The conflict in this story can also be viewed as internal, as Montresor describes each of his decisions, but through his actions he shows his internal conflict to be inconsequential. It is arguable that he has no internal conflict nor any regrets. This lack of feeling is part of what makes the story so creepy.