How does "The Tell-Tale Heart" reflect that Edgar Allan Poe was mentally ill?

We need to be very careful with our analysis of this story
and the links that we make between it and the author. The unwritten assumption in your
question is that Poe was actually mentally ill, whereas we have no such proof. Some
critics argue that because of the convincing descriptions of insane first person
narrators, Poe himself must have suffered some sort of mental illness because of the
understanding and insight he shows in creating such characters. This story is certainly
one of these narratives, as the anonymous first person narrator is shown to be
distinctly unreliable because of his insanity whilst at the same time he protests that
he is actually well and not mentally ill. Note how this is achieved from the very first
paragraph of the story:

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True!--nervous--very, very dreadfully nervous I
had been and am; but why will you say that I am mad? The disease had sharpened my
senses--not destroyed--not dulled them. Above all was the sense of hearing acute. I
heard all things in the heaven and in the earth. I heard many things in hell. How, then,
am I mad? Hearken! and observe how healthily--how calmly I can tell you the whole
story.

Note how the narrator
juxtaposes the claim that he can hear "all things in heaven and earth" and "many things
in hell" with the question "How, then, am I mad?" Such juxtaposition reinforces the
insanity of the narrator. However, the fact that Poe is able to create such convincingly
insane characters does not therefore indicate that he was insane
himself.