The old man's eye is blue with a "film" or "veil" covering it. This could be a medical condition, like a corneal ulcer, but symbolically it means that the characters have issues with their "inner vision" – what's commonly known as one's outlook on the world. They are stuck. Everything is obscured for them. Our reading of the story is likewise filtered through this hazy eye, causing at least some confusion and frustration with the text.
The eye also does some pretty weird stuff. It seems dull and unseeing – yet, it has strange powers. It makes the narrator's blood run cold. It "chill[s] the very marrow in [his] bones" (6). After hiding the old man's body, the narrator "replaced the boards so cleverly, so cunningly, that no human eye – not even his [the old man's] – could have detected any thing wrong" (8). Interesting. That statement implies that at some point the eye could see hidden or secret things.
The eye also seems to have a bodyguard, the heart. When the narrator trains the beam on the open eye, it causes the heart to beat an alert. When the policemen are there, the heart beats loudly to alert the cops – so the eye can again see and be seen.
The narrator is fixated on the "vulture eye" aspect of the old man's eye. He brings it up three times. Vultures prey on the sick or dead, and they gorge themselves to the point of stupor. Whether or not the old man is a vulture-like person, we can't know. But that's what he symbolizes to the narrator. If vultures prey on the dead and almost dead, and the narrator is afraid of the "vulture eye," does this mean the narrator is dead or almost dead?