There are several possibilities. Either the narrator hears, or thinks he hears, the heart of the old man he has murdered; or it is his own heart, or his own pulse, he hears beating in his ears; or thirdly there is nothing at all to hear and he only imagines he is hearing a heart beating.
He cannot be hearing the old man's heart because the old man is dead. Poe makes it clear in the following paragraph that the old man could not possibly be alive.
If still you think me mad, you will think so no longer when I describe the wise precautions I took for the concealment of the body. The night waned, and I worked hastily, but in silence. First of all I dismembered the corpse. I cut off the head and the arms and the legs.
And if the victim cannot be alive, his heart cannot be beating. It would seem that Poe does not want his reader to get the impression that his narrator really hears the victim's heart, even though Poe has his narrator say that it is the old man's heartbeat he is hearing up to the time he confesses to the murder.
So the narrator must either be hearing his own pulse beating in his ears and imagining that it is the victim's heartbeat, or else there is nothing in the nature of a heartbeat to be heard, and the narrator is just imagining he hears the pulsating sounds he describes.
The opening sentences of the story seem intended to create the impression that it is his own pulse that finally drives the narrator to break down and confess.
TRUE!—NERVOUS—VERY, very dreadfully nervous I had been and am; but why will you say that I am mad? The disease had sharpened my senses—not destroyed—not dulled them. Above all was the sense of hearing acute. I heard all things in the heaven and in the earth.
He is VERY NERVOUS, but he is trying to act perfectly calm. The fact that the heartbeat keeps growing louder and faster strongly suggests that it must be his own pulse he is hearing. If the sound were completely imaginary and had no source in reality, then it seems most likely that he would imagine the volume and tempo to remain the same. The sound can only keep increasing in volume and tempo if it is his own heartbeat stimulated by guilt and fear that is causing the tell-tale sound. It must be the narrator's own heart.
Poe's story of psychological terror may well have a double entendre for a title. That is, the "tell-tale heart," or tattle-tell heart, can be either the narrator's or the old man's heart since the teller of the tale is certainly an unreliable narrator.
That the narrator is unreliable is indicated by his first lines, suggestive of mental imbalance.
True! nervous, very, very, dreadfully nervous I had been and am, but why will you say that I am mad?
His psychological fear of implacable time and death, represented to him by the old man and, especially, his "vulture eye," become an obsessive terror. In order to rid himself of his horrendous fear, the narrator tries to eradicate the eye, and he rushes into the room, pulling the bed over the old man. "But for many minutes the heart beat on with a muffled sound," the narrator declares.
This declaration is the admission that the narrator hears the man's heart in his mind. Or, it may be his own heart beating nervously and his blood pressure pounding in his ears. For, after the police officers who enter because someone heard a scream and linger on, the narrator can bear their presence no longer, exclaiming,
"Villains!....dissemble no more! I admit the deed!--tear up the planks!--here!here!--it is the beating of his hideous heart!"
In his excited stated--"very, very, dreadfully nervous"--the narrator's heart beats rapidly; in addition, his raised blood pressure, that pounds in his ears, makes him believe that he "hears" the pounding of the old man's heart, as well.