For a three-act plot analysis, put on your screenwriter’s hat. Moviemakers know the formula well: at the end of Act One, the main character is drawn in completely to a conflict. During Act Two, she is farthest away from her goals. At the end of Act Three, the story is resolved.
Act I
The narrator wants to prove his sanity by telling us a story from his past. The first act of that story deals with eight nights in the life of the narrator. The first seven nights are pretty much the same, and consist of the narrator peeking into the old man's bedroom while he sleeps, hoping he'll open his "Evil Eye" so the narrator can kill him. On the eighth night the man's eye opens.
Act II
At the beginning of the second act, the narrator jumps into the room, kills the man with his own bed, cuts up the body, and hides it under the floor of the man's room.
Act III
The narrator doesn't even get a chance to rest before the police show up. Act three consists of the narrator at first acting cool, but quickly disintegrating due to a mysterious ticking sound that just keeps getting louder. When the narrator can't take the sound anymore, he gives up and points the cops to the body, announcing that the noise is the heartbeat of the murdered old man.