The reference to a vulture's eye is found early in "The Tell-Tale Heart" by Edgar Allan Poe. The narrator tells us that he is nervous, mad (insane), upset--and yet quite calm. He starts to explain his story and what prompted him to do something dramatic and, probably, unforgivably bad.
He tells us about "the old man," someone the narrator both claims to love and wants to destroy. These two emotions are not usually compatible; however, in this case, the narrator thinks he can justify his behavior by explaining something important about the man.
He had the eye of a vulture--a pale blue eye, with a film over it. Whenever it fell upon me, my blood ran cold; and so by degrees--very gradually--I made up my mind to take the life of the old man, and thus rid myself of the eye forever.
The narrator did not want the man's money, has not somehow been hurt by the man; in fact, he claims to have a greta deal of love for the man. Somehow he thinks that this man's eye outweighs everything else about who the man is and how much he cares for the man.
Poe is a master of the grotesque image, and a vulture's eye seems to suit that imagery quite well. It is not "normal" because it has a film of some kind over it, and the image implies some kind of evil. Several possible interpretations are possible.
Since the narrator refers to the vulture eye three times, it does seem to be a kind of fixation for him. It also seems to have some sense of power, at least in the mind of the narrator. This eye is able to "chill the very marrow in [his] bones" and "see" things that others cannot see (though even the old man's eye cannot see where the old man's body is now). And yet the eye was not able or powerful enough to warn the man of his own imminent death.
Vultures are creatures which prey on the dead, so perhaps the vulture eye is a kind of foreshadowing of the narrator's death.
The word "eye" is a homonym (sound-alike word) for "I," and it is possible that our mentally disturbed narrator actually wanted to kill himself when he killed the "eye." He tells us he wants to kill the old man and his eye, but perhaps he really just wants is to kill himself. This is a reasonable ironic theory, given Poe's use of irony in his writing. For example, in this story, the narrator is quite afraid the neighbors will hear the old man's loudly beating heart (which obviously no one but him hears); so he screams as he kills the old man and we know they heard that. This play homophonic play on words is something Poe would do, and the eNotes "themes" link, below, discusses this idea further.
Whatever else it is, the old man's eye is not literally like a vulture's eye. That means there is some symbolic meaning, and there are several possibilities from which to choose. Whatever it is, the narrator seems to be the only one moved by it.