The narrator of "The Tell-Tale Heart" is unreliable. He is trying from the very beginning of the story to make a case for his sanity, but the very story he tells completely undermines and is at odds with his assertions of sanity. Throughout his narrative, he recalls the events that led him to murder the old man and then confess. He attempts to illustrate his clearheaded reasonableness with examples. For instance, he tells us that he "loved the old man" and that the old man had never wronged him. This admission makes his murder of the man less, rather than more, understandable or sane. Why would you murder someone you loved who had done you know wrong—and merely over his eye, as the narrator insists is the case?
He also asserts that he cannot be mad for the following reason:
Now this is the point. You fancy me mad. Madmen know nothing. But you should have seen me. You should have seen how wisely I proceeded—with what caution—with what foresight—with what dissimulation I went to work!
Yet, we know that madmen can be quite knowledgeable—and can proceed with caution and foresight. The narrator completely misses the point while thinking he is making a valid point: it is not how he proceeded but what he did—murdering an innocent man for no good reason—that attests to his lack of mental stability.
Finally, he hears the dead man's heart beating at the end, probably a manifestation of his own guilt. Yet, rather than assume he might be hearing things or hallucinating, the narrator violently insists that the police are mocking him:
They heard! —they suspected! —they knew! —they were making a mockery of my horror!—this I thought, and this I think.
The disjunction between what happens and how the narrator interprets it leads us to conclude he is unreliable.