In "The Tell-Tale Heart," the narrator clarifies that he is nervous, not mad, because he wants to make his story sound credible.
He knows that if anyone thought for one moment that he was mad, then they wouldn't believe a word of his story. It would be regarded as nothing more than the ravings of a madman, easily ignored and just as easily rejected.
The narrator presumes that his audience already thinks he's mad. After all, the hideous crime he's committed, the senseless murder of an old man, would seem to indicate insanity on his part. That being the case, it's all the more important for the narrator to go out of his way to convince his audience that, contrary to what they might think, he really isn't mad after all.
Claiming that he's nervous rather than mad also allows the narrator to account for some rather bizarre details concerning the murder. For instance, he claims that it is his nervousness that's responsible for his acute hearing. In turn, it's supposedly because of this that he's able to hear the beating heart of the old man even after he killed him.
However, no one could seriously be convinced of the narrator's efforts to make himself out to be nervous rather than mad. No matter how acute one's hearing may be, it would be impossible to hear the dead man's heart beating at this late stage.