This is a great question given that any given reader could justify their own personal thoughts on the subject in many different ways.
As for me, I believe that The Tell-Tale Heart is a first-person narration where the narrator is speaking directly to me.
TRUE! nervous, very, very dreadfully nervous I had been and am; but why WILL you say that I am mad?
The use of the personal pronoun "you" tends to make me feel as if the speaker is telling me his story in an effort to prove to me that he is not mad. Throughout the entire story, the speaker seems to desire my approval of his actions and an acceptance of his mental stability.
While this may not be the case for every reader, one could easily justify that the narrator is speaking directly to each, and every reader, simply based upon the fact that he is addressing the reader through the use of "you."
Outside of that, one could also justify that the story is a retelling of the narrator's story by the police or other person to whom he had spoken to regarding his past. It could easily be a doctor (psychiatrist, psychologist). It simply depends upon the reader's individual perspective and justifications made based upon textual evidence.