The narrator feels the fear that he will be pronounced insane, followed by the fear that he will be caught. He tells the reader multiple times at the beginning of the story that he is not insane and even gives his own version of proof—would an insane man go to such great lengths to hide the body, being careful not to even spill any blood? The narrator is also afraid that he will be exposed by a heart that everyone should be able to hear, but apparently he is the only one that can hear it. Thus, he rationalizes this as his own acute sense of hearing but he does not make any mention of other sounds that he hears. He is trying to rationalize his guilt.
He even justifies his action at the beginning of the story by talking about the old man's vulture-like eye, and by explaining that the eye is evil, the narrator justifies killing the old man. He even goes into detail to explain that he did not kill the man for material gain or out of hatred for him personally.
In Poe's story, The Tell-Tale Heart, the narrator fears several things. On the surface, he fears the old man especially because of the eye. When the narrator can stand the eye no longer, he kills the old man and buries him beneath the floor. Then his fear becomes one of discovery because if the old man's body is discovered, the narrator will be arrested. At the core of the story, however, is the fear that he will be considered insane. That is why he makes reference to proving that he cannot be considered insane. The title also refers to the heart which will tell the tale of the murder. Hearing the heart beating with his guilt when no one else can hear it, the narrator's insanity becomes clear to the reader and reveals the narrator's greatest fear.