The narrator is in fact insane. Because he uses specific details in the story. Near in the middle of the story he describes a sense of right after killing the old man. He said “I then smiled gaily, to find the deed so far done. But for many minutes the heartbeat on the muffled sound. This, however, did not vex me; it would not be heard through the wall.” Also he tries to convince us that he is not insane.
Through the narrators actions and dialogues, you can see that he is indeed insane. The narrator is also unreliable in the story. He can still hear a dead man’s heartbeat. That is how you can tell that this narrator has in fact lost his mind
Hope this helps