"The Tell-Tale Heart" has certain satirical elements, in the sense that it contains the satirical technique of irony. The story starts out with the speaker's repeated professions of sanity, followed by a very insane tale. Driven mad by his older roommate's eye, the speaker relates his very detailed plan for murdering the roommate in his sleep. This plan includes several trial runs, designed to accustom himself to the dark and to not arouse his roommate's suspicion. On the chosen night, the speaker is successful in smothering his roommate and hides his lifeless body beneath the floorboards in his room; however, a neighbor has heard sounds of a struggle and the police arrive to question him. The speaker almost brags about his coolness under pressure and the ease with which he fools the investigators until he begins to hear a muffled sound, almost like a heartbeat, under the floorboards. As the sound becomes seemingly louder and louder, the speaker loses his composure and eventually confesses to the crime.
One message the story could be conveying is the thought that "the best laid plans of mice and men often go awry." No matter how meticulously the speaker planned the murder of his roommate, it was ultimately his own conscience or lack of self control that led to his confession and arrest.
Another potential message is the old adage that an insane person does not recognize that he or she is insane. The speaker starts by declaring repeatedly that he is not insane, and begs the reader to not judge him so. Then, he proceeds to tell a story of a sincerely insane man, driven mad by an insignificant physical oddity, who confesses to murder after hearing phantom sounds.
Since the purpose of satire is to point out some flaw in the text's subject and, perhaps, inspire some change relative to that flaw, it seems possible that Poe is emphasizing the common fear of death and the lengths to which maintaining such an irrational fear can drive us. Many of us don't really like to consider our own deaths, and the narrator is no exception. When the old man's "vulture eye" begins to remind him of his own mortality, he desperately needs to get rid of it. The eye likely seems "veiled" because the old man has cataracts (a malady associated with old age, and those in old age are nearer to death), and the narrator's description of the eye further links it to death because vultures are associated with death as well.
The narrator insists, again and again, that he is not mad, but it becomes clearer and clearer to the reader that he very much is mad, or at least that he's been driven mad by his need to escape reminders of death. Therefore, since we can have no effect on our own mortality, and we cannot prevent death from "approaching" and casting "his black shadow" over us, it really does no good to dwell on it. In fact, it can do irreparable harm to us if we do.