Perhaps the vulture eye is an omen, reflective of what is to happen to the narrator. For, the narrator declares, "Whenever it fell upon me my blood ran cold." And, so, the narrator becomes disturbed in a cold premonition of evil to come. Thus, he creates a rational for killing the old man whom, he declares, he loves. He must be rid of the eye to which he ascribes evil: "For it was not the old man who vexed me, but his Evil Eye."
In his stories Poe applied a technique which he termed "arabesque." This arabesque is a twisting and turning of details in a horrific way. In "The Tell-Tale Heart," it is, indeed, the heart that tattles on the evil-doer. However, rather than being the heart of the victim as the narrator imagines, it is the beating of his own heart that the narrator hears, his own conscience which he tries to silence long before the murder. This guilt is what causes the narrator's "blood to run cold" in the beginning when he sees the vulture's eye. For, in an arabesque, the narrator sees already reflected in the eye the murder which he will later commit. That is, he sees in the vulture eye of death, the evil which his soul already knows before the deed.
A vulture could be a symbol of coming death. In the wild, when a vulture spots a sick or injured animal, it starts circling, waiting for it to die, and waiting for its chance to feed on the remains. So, for the narrator to compare the old man's eye to a vulture, it could represent the coming death of the old man, and how his time is soon over.
Also, the narrator himself could have felt the prey of the old man. A vulture is very a watchful, threatening bird. It isn't a pretty songbird or a harmless robin; a vulture is large, predatory, and menacing. Perhaps the narrator felt himself the potential victim of the old man, being stalked and pinioned for some evil plan. He felt he couldn't escape that evil, watchful eye, so decided to strike first.
Plus, a vulture's eye is just a creepy image; we can't discount Poe's proclivity for picking his images very carefully to enhance the overall mood of the piece.
The narrator describes the eye as being like the eye of a vulture. Vultures are scavangers that swoop in upon dead animals to feast on the carcass. Thus, they are ever-present and diligent. They see everything.
The eye in this story symbolizes that sort of penetration. The narrator is obviously disturbed and terrified that someone will see into his deepest fears and violent plans. The eye represents the window into the mind and soul of the narrator. It is always watching him - no matter what, he will be observed. This comes true in the end when the police come and the narrator reveals his own guilt. Somone is always watching - someone always knows.
You could also say that the eye is the conscience of the narrator. It watches him because it knows that he is planning to do wrong, and the narrator is thus haunted by it. Instead of trying to reconcile his desire with what his conscience knows is right, he chooses to destroy it.
The narrator also refers to the eye as 'a pale blue eye, with a film over it' suggesting that it has a cataract. New World vultures are classed as birds of prey as well as scavengers and the narrator may have been considering that the old man may be either predator or prey himself - with 'vulture' tendencies he could be the hunter or the hunted.
Note that the narrator refers to the old man's eye as an "Evil Eye," and we associate vultures as creatures of evil due to their rather antisocial nature of eating dead animals and humans. This helps us therefore to understand the profound impact that the eye had on the narrator and why it haunted him so.
The eye is a malignant source of evil, and the word "vulture" is used because the vulture is black (black = evil), and it hovers about, waiting to feast on dead flesh.
What I got from it was that perhaps the "vulture eye" could be the someone who judges him. Like the hunter who preys on someone and feels of them (Edgar's writing, style of life, etc.)
Te eye in the story is obviously described as horrible and 'the eye of a vulture.' A vulture is a bird of prey and we usually associate those with death. A vulture also flys high over things, always watching so we could assume that the old man is a parental type character, perhaps a father or a master to the narrator - although we can never be sure. The old man doesn't like being always watched over so he decides to destroy the old man to be rid of this feeling of being trapped by it. However in the end when the police catch up with the narrator it is proven that someone is always watching you.
The eye could also represent a theme for the story - The old man has a hideous, repulsive eye; outwardly, he is ugly. But, as the narrator admits, he is otherwise a harmless, well-meaning person. The narrator, on the other hand, is inwardly ugly and repulsive, for he plans and executes murder; his soul is more repulsive than the old man's eye. For this reason he is a worse person and in the end doesn't get away with what he has done.