It may seem strange to think of the narrator of "The Tell Tale Heart" -- a murderer -- in a sympathetic light, but to an extent Poe is able to make him so through strong characterization and point of view. Since the whole story is told from the point of view of the first person narrator, we as readers internalize the narrator's mental illness. Poe achieves this with remarkable economy: the very first lines of the story rope us in:
It’s true! yes, i have been ill, very ill. But why do you say that I have lost control of my mind, why do you say that I am mad?
"We" haven't said anything yet, but it is clear that, despite his protests, our narrator is quite crazy. Even though what the narrator says makes no sense -- there is no "reason" why the old man's vulture eye should cause the narrator to murder him -- we nevertheless feel his paranoia and fear with him. We know the old man to be guiltless, but we understand the irrational source of the narrator's fear. When the police come, real suspense is generated: we don't want the police to find the narrator out, and are distressed when the narrator confesses at the end of the story.
For more on "The Tell Tale Heart," and Poe's aesthetics, see this eNote.