As the narrator mentions, he has no explicit reason for killing the old man. No relationship between them is mentioned; he is possibly the old man's son or nephew, or a live-in caretaker. Of course, it must be remembered that the narrator is unreliable; despite his protestations to the contrary, he is clearly insane to some degree, and possibly also delusional (one interpretation is that he never killed the old man at all, and that is why there is no blood). In the opening, he states:
It is impossible to say how first the idea entered my brain; but once conceived, it haunted me day and night. Object there was none. Passion there was none. I loved the old man. He had never wronged me. He had never given me insult. For his gold I had no desire. I think it was his eye! yes, it was this!
(Poe, "The Tell-Tale Heart")
The narrator is suffering from an extreme version of morbid irritation; he finds the presence of the old man's "vulture eye" intolerable, and instead of searching for other employment or other living circumstances, he decides that the only rational solution is to murder the old man, thus ridding himself of the "vulture eye." His focus is on the eye itself, as he is unable to kill the old man until the eye is open and visible. However, considering how unreliable he is, it is impossible to say that greed or some other justification is not responsible for his actions.