In Edgar Allan Poe's "The Tell-Tale Heart," the conflict between the narrator and the old man primarily arises from the narrator's intense irrational obsession with the old man's eye, which he describes as "vulture-like." The narrator feels a strong aversion toward the eye, which he perceives as evil. This obsession drives him to a point where he believes he must rid himself of the old man in order to free himself from the torment of the eye.
The internal conflict is rooted in the narrator's struggle with his own sanity, as he insists he is not mad, despite his violent thoughts and actions. This psychological turmoil escalates into an external conflict when he ultimately decides to kill the old man in a misguided attempt to eliminate the source of his distress. The narrator’s attempt to justify his actions and maintain control over his emotions hints at deeper themes of guilt and madness that permeate the story.