There could be many different answers to this question. I for one, think that the dilemna is the narrator. He insists he isn't mad, that his disease has harkened his senses. The reader, of course, thinks they know that he is mad by his reasoning for killing the older man. But, do we ever truly know? No, because the narrator tells the story first person, so the reader never truly knows how much of what he is saying is true. Since the story has no outside influences, it is impossible to know that is/has really gone on. Mad or not, the narrator can never truly tell the reader what he/she wants to know, which is the very literary device Poe uses to make this story such a psychological thriller.
When I teaching my students "The Tell-Tale Heart", we discuss the guilt the narrator feels over killing the old man. He fully admits in the beginning of the story that is fond of the old man. Also, the narrator's own terror and awe is related to his obsession with time. He speaks of spending hours at the old man's door, talks about the the time it took to bury the old man. He is mad in his obsession with time and the eye, and guilty becuase of the the murder. There are in essence, two dilemmas.