The opening of the story gives one example-he openly admits to having a "disease" that "sharpened [his] senses" and openly admits to being a "very, very dreadfully nervous" person.
He also seems a bit preoccupied with the old man's "vulture" eye; this eye so bothers him that he decides to take the old man's life "and thus rid myself of the eye forever".
Once he finally succeeds in killing the man, he admits to still hearing that "hideous heart" still beating, even after the man is dead; this, of course, happens at the end of the story also when the police are there to investigate-the narrator can hear the sound of the heart under the floor and acts quite peculiarly: "They heard! They suspected! They knew! They were making a mockery of my horror!"
The narrator definitely shows traits of being "insane": he has unusual preoccupations, he plots a murder and succeeds in its execution, he shows manic behavior patterns, and he is delusions of persecution (that others are somehow out to get him).