The narrator mentions a "watch" four times in the story. A watch is a visual and auditory representation of time. The watch watches time, and tells tales of time. Time can also be said to be watching death, up ahead in the distance. Each tick of the watch symbolizes a movement closer to the inevitable death that all humans face. Poe presents this subtly in the story's first mention of the watch: "A watch's minute hand moves more quickly than did mine" (4).
This of course is on the eighth night. Here the narrator compares himself to a watch, a watch watching the old man's death. The narrator steals time's power as an agent of death. The narrator literally controls the time of the old man's death. He's a walking "death watch."
This metaphor/word play becomes more explicit in the second mention of time in the story: "He was still sitting up in the bed listening; – just as I have done, night after night, hearkening to the death watches in the wall" (4).
This is a mystifying line, until we know that "death watches" are kind of beetle. Death watch beetles live inside walls, and bang their heads on said walls to attract mates (source). Poe might not have known this was a mating call, and was likely referring only to the popular belief that the banging is a countdown to someone's death. Then again, maybe he did know. Intentionally or not, this odd moment in the novel juxtaposes sex and death in a way that would have made Sigmund Freud proud. In Beyond the Pleasure Principle Freud theorized that death and sex are intimately and intricately intertwined.
Now for the second and third mentions of "watch" in the story:
[N]ow, I say, there came to my ears a low, dull, quick sound, such as a watch makes when enveloped in cotton. (6)
It was a low, dull, quick sound – much such a sound as a watch makes when enveloped in cotton. (9)
The old man's heart is also a watch, as we see in these almost identical passages. It both watches and counts down the time leading up to the man's death (first passage). Then the heart/clock becomes a zombie of sorts (second passage). It resurrects itself so it can tell about the time (of death) it watched, in a sense taking time back from the narrator. Pretty clever.