The most reasonable answer to this question depends upon the time period in which the narrator becomes a defendant because the political temper of an era affects how decisions about prisoners are made. During the time period of Poe's writing of the story, for instance, The County Asylums Act of 1808 had established institutions for the criminally insane and allowed for treatment rather than just incarceration. Usually, the death penalty was not given to a person who was found to be insane.
Clearly, Poe's narrator seems quite mentally unstable. Initially, he calls himself "nervous," but not mad. And, his rationale for committing the murder seems rather unnatural and irrational, to say the least:
I loved the old man. He had never wronged me.... For his gold I had no desire. I think it was his eye! yes, it was this! He had the eye of a vulture—a pale blue eye, with a film over it.
The fixation on and the depiction of the old man's eye indicates that the narrator is not a rational man. And, of course, the calm recounting of how he dismembered the body and buried it under the floor boards, is certainly not normal. But, the horror that exists inside the narrator's own mind is, perhaps, the most convincing argument for a verdict of insanity. For, after he has succeeded in his crime, the narrator is horrified by his own action and imagines that the policemen can hear the heartbeat of the old man whom he has killed, when, in truth, it is probably his own nervous heart. These wild imaginings, therefore, are what convict the murderer.
In Edgar Allan Poe's "The Tell-Tale Heart", there is no question that the narrator committed murder, for we have an unambiguous confession. We can also deduce from the circumstances that the officers could find unambiguous forensic evidence to substantiate the claim that the narrator did, in fact, commit the murder. Moreover, there is no chance that the murder was in self-defense or an accident -- the narrator was fuilly aware that he was killing the old man. However, there is some question as to whether the narrator is sane; at the end of the story, the narrator describes himself as losing even the surface imitation of sanity he tried to display earlier:
I gasped for breath, and yet the officers heard it not. I talked more quickly, more vehemently but the noise steadily increased. I arose and argued about trifles, in a high key and with violent gesticulations; but the noise steadily increased. Why WOULD they not be gone? I paced the floor to and fro with heavy strides, as if excited to fury by the observations of the men, but the noise steadily increased. O God! what COULD I do? I foamed -- I raved -- I swore! I swung the chair upon which I had been sitting, and grated it upon the boards, but the noise arose over all and continually increased. It grew louder -- louder -- louder! And still the men chatted pleasantly , and smiled. Was it possible they heard not? Almighty God! -- no, no? They heard! -- they suspected! -- they KNEW! -- they were making a mockery of my horror! -- this I thought, and this I think. But anything was better than this agony! Anything was more tolerable than this derision! I could bear those hypocritical smiles no longer! I felt that I must scream or die! -- and now -- again -- hark! louder! louder! louder! LOUDER! --
Thus a judge should sentence him not guilty by reason of insanity and send him to an appropriate medical institution.