The central conflict in this story exists between the narrator and nature. It is death that he fears and tries to avoid, even though death is natural. He admits that he has no motive for killing the old man other than the man's "vulture eye": he doesn't hate the old man or want his money; in fact, he says that he loves the old man. Vultures are often associated with death because they are scavengers that feed on carcasses. Furthermore, old age is associated with death as well, and so we might interpret these overlapping associations as evidence to suggest that the narrator really only wants to kill the old man because he reminds the narrator of death, of his own mortality, and this is a frightening thought.
Moreover, we don't see a conflict within the narrator: he is intent on killing the old man and does not seem to doubt or question this decision. He even remarks, on the night he actually murders the old man,
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.
The narrator refers to deathwatch beetles, as superstition states that one hears the clicking noises they make before someone dies. Here, the narrator admits that he is often kept up late at night listening for these sounds, perhaps in fear for himself. Further, he describes the fearful sound the old man makes as a groan "that arises from the bottom of the soul when overcharged with awe," saying that "Many a night," this same groan "has welled up from [his] own bosom." It is death that he fears, it is death with which he does battle, and his fear drives the conflict in the story.
The conflict is never really resolved. The narrator kills the old man, and this does nothing to make the narrator feel better. For a moment, he seems to, but then when the police officers arrive, he is again beset by his nerves. The conflict continues because he is not yet dead, and so his fear of death remains.