Why did the narrator of "The Tell-Tale Heart" kill the old man?

Poe's classic short story "The Tell-Tale Heart" is told by an unreliable narrator, who is clearly mentally unstable. There are numerous clues that suggest the unreliable narrator is insane, which include his continual insistence that he is not "mad," his staccato writing style, and his "sharpened" senses that allow him to hear paranormal sounds. Toward the beginning of the story, the narrator mentions that he had no ill will toward the old man and actually loved him—another example of his insanity. After reading this statement, the reader cannot help but wonder how someone could murder a person they love. The narrator goes on to say that he was not motivated by money to kill the old man and was simply disturbed by the old man's "Evil eye." The insane narrator explains his reason for killing the old man by saying,

I think it was his eye! yes, it was this! 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. (Poe, 1)

One could argue that the narrator's desire to murder the old man stems from his mental instability. He is clearly mentally ill, and his motivation to kill the old man is not a strong enough reason to commit such a horrific crime, in the eyes of most people. Therefore, one could surmise that the narrator murdered the old man because he was mentally ill and insane.

To answer this question, take a look at the second paragraph of the story. According to the narrator, the old man had an "evil eye" which made him feel very uneasy:

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.

In other words, the narrator felt compelled to kill the old man because this was the only way that he could stop the evil eye from bothering him. As a result, he immediately made plans to murder the old man.
However, if we look a little deeper, it becomes clear that the narrator killed the old man for another reason. Specifically, we can argue that the narrator killed the old man because the narrator was not of sound mind. In the week preceding the murder, he was clearly suffering from some kind of mental illness which caused a deep sense of paranoia about the old man. Although the narrator opens the story by claiming that he is not mad, his motivation for the murder suggests otherwise.