As the Invisible Man, in Ralph Ellison's Invisible Man says, "The end [is] in the beginning." As in that novel, chronologically speaking, the end of "The Tell-Tale Heart" actually takes place before the beginning. We can look at this in several ways.
First, notice the structure of the narrator's tale. It's completely linear, following the narrator's activities through eight nights. The beginning (which is really the ending) is a continuation of this linear narrative, but perhaps it's a more immediate continuation than we might have thought. Look how it seems to start in the middle of a previous conversation with someone:
True! – nervous – very, very dreadfully nervous I had been and am; but why will you say that I am mad? (1).
Maybe this is what the narrator tells the police after he shows them the body. Maybe this is part of the interrogation process, which is followed by the story of the murder.
Or, maybe much more time passes before the narrator tells the tale. Suppose he was sentenced to do his time in a psychiatric facility. He's been in a number of years, and it's possible he'll be released, if he can prove his sanity. So he tells the story to demonstrate that he's ready to face the outside word. Needless to say, the story would have the opposite of the intended effect in this situation.
Some interesting things also occur in the literal ending of the story, by which we mean these lines:
"Villains!" I shrieked, "dissemble no more! I admit the deed! – tear up the planks! here, here! – It is the beating of his hideous heart!" (10)
First, the narrator calls the police "villains," and accuses them of "dissembling." Dissembling is pretty close to "dissimulation" (see paragraph 3). Dissemblers and dissimulators both act one way in order to conceal true feelings, or intentions. If you recall, the narrator cites his ability to act sweetly to the old man while inwardly desiring to kill him as proof of his sanity. Now he suspects the police of doing the same thing – acting like they don't suspect him, even though they do.
The narrator might well be correct in this, though what likely made the police suspect him was not that they could hear what hears, but his own actions, specifically, this:
I foamed – I raved – I swore! I swung the chair upon which I had been sitting, and grated it upon the boards. (9)
In any case, by calling the police "villains" for acting one way and feeling another, the narrator admits (in a round about way) that he too is a villain. By connecting his auditory hallucinations with the old man's heart, he admits he actually feels bad about what he did, or at least knows it's wrong. That sounds something like sanity, which might explain why the narrator would end the story meant to prove his sanity with what, at first glance, looks like a confession to murder.