This excellent story, like "The Black Cat," shows an unreliable narrator who gives himself away because of his own unbalanced nature. For me, a key quote to consider in how Poe uses the heart beat of the old man to show the mental instability of the narrator is the first mention of the heart-beat:
And have I not told you that what you mistake for madness is but over-acuteness of the sense? - now, I say, there came to my ears a low, dull, quick sound, such as a watch makes when enveloped in cotton. I knew that sound well, too. It was the beating of the old man's heart. It increased my fury, as the beating of a drum stimulates the solder into courage.
Of course, because of the character of the narrator, we disagree with his first sentence. His "hearing" of the heart beat reflects his own inner mental instability. It is interesting how, once the man is dead and disposed with, that the beating begins again, just at the time when it looks as if the narrator is going to be successful in getting away with the crime. The beating of the heart then could be said to describe both the madness of the narrator but also his guilty conscience at some kind of level, representing the guilt he feels about killing the man.