In "The Tell-Tale Heart, why does the narrator kill the old man?

Personally, I'd suggest that, regardless of his words to the contrary, the main character of "The Telltale Heart" is of a deeply deranged personality. This is encapsulated in the story's opening paragraph, and it carries across the tone of the entire story that follows. For all that the narrator claims to be sane, one must ask whether that claim matches reality as the story depicts it.
In any case, the narrator gives his explanation as to why he committed the murder. He claims that this act was not driven out of hatred or greed. Rather, he was driven by disgust as to the appearance of the old man's eye (which is described like a cataract, and which the narrator explicitly compares to a vulture's eye). He voices a deep loathing of it, and it is out of this loathing that he commits the murder.