The narrator is indeed insane, which makes him an unreliable narrator. The narrator begins the story by saying ‘‘True!—nervous—very, very dreadfully nervous I had been and am, but why will you say that I am mad?'' We know he is crazy 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.
All the signs that point to his insanity and instability also clue us to the fact that he is an unreliable narrator. He does not see things the way a normal person would and distorts reality to suit his needs. This is the epitome of that an unreliable narrator is.