I love the idea that one of the previous posts suggested referring to the "death watch beetles" that may have been mating in the walls of the house, thus making the ticking sound. (The poster was incorrect about the narrator mentioning this possibility in the story, however.)
The real question is whether the narrator actually hears a sound at all, or whether he simply imagines it. The fact that it is "such a sound as a watch makes when enveloped in cotton" suggests the possibility of a watch or clock. However, none is mentioned in the story (unlike the magnified ticking of the watch heard by the doomed man in Ambrose Bierce's "An Occurrence at Owl Creek Bridge.") Since the police officers hear no sound, it can only be assumed that it is the guilty conscience of the madman. Only one other possibility exists: Having already admitted that he has a
... sense of hearing acute. I heard all things in the heaven and in the earth. I heard many things in hell...
it could be that the narrator's ears are indeed attuned to the underworld, where the time for his arrival is slowly ticking away.
Poe's stories are very dark. They often deal with death and insanity. I would say that the "heartbeat" is the narrators insanity. He starts off believing that the old man must die because of his "vulture eye." This insanity transcends after the narrator has killed the old man and buried him in the floor to the never-ending heartbeat that he hears after he murders the old man. When Poe was writing his readers would have been especially interested in the insanity defense (1840) and this interest could have been Poe's purpose in making the narrator seem insane. The narrator also talks about the old man being near death anyway, thereby making his murder more of a mercy killing. Another idea is that at the time superstition said that there were "death watch beetles." The narrator first admits to hearing death watches in the wall after startling the old man from his sleep. According to superstition, death watches are a sign of impending death. One variety of death watch beetles raps its head against surfaces, presumably as part of a mating ritual, while others emit a ticking sound. So this could have been what Poe was going for. Personally I like the insanity explanation better. I hope this answer was helpful.
I believe there are numerousl answers to this question. One that comes to mind is the feelings you would get when you were a kid if you did or were about to do something wrong. When your adrenaline is pumping and you are scared, all you hear is your own heart beat. I know the narrator is unreliable and mentally unstable; however, the heart beat is his own as he believes he has killed the man and everyone else is aware of that dark detail.
the "heartbeat" you are talking is cause by the narrators insanity, and maybe in the deeper aspect of Poe's life, this story reflects the authors personal life. because of the tribulations that happen to Edgars life his way of living beacame that lowly hearted and low in spirit.