The narrator in "The Tell-Tale Heart" is a murderer. What are examples of his insanity?

The narrator is indeed mad, which also makes him an unreliable narrator. We know he is mad because he has no real reason to murder the old, defenseless man other than the fact that he has a peculiar eye. The palish-blue eye taunts and irks the narrator to the point of insanity. The man decides to murder the bearer of the eye but cannot do so when his eyes are closed. The insane narrator rationalizes that his crime must be committed when the eye is in view because he is in fact antagonized by the eye and not by the old man. This rationalization, although perfectly normal to the story teller, makes no sense to a sane person.

For eight nights, the madman stalks his victim as he meticulously plans his crime. Ironically, this carefulness to him is a sign of his sanity, and he assures the reader that no madman would be able to be so keen. However, the reader sees the man's reasoning of the ploy as utterly lunatic, and we watch the narrator on his path through madness.

Once the murder has been committed, the narrator demonstrates his insanity once again by imagining the sound of the victim's heart. It beats louder and louder until he finally confesses.