I would say that the narrator's conflict is completely internal in Edgar Allan Poe's "The Tell-Tale Heart." He admits from the very beginning that there is no specific reason for his actions.
Object there was none. Passion there was none. I loved the old man. He had never wronged me. He had never given me insult. For his gold I had no desire.
The only excuse given for the plotting of the old man's murder was the appearance of his eye.
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. Whenever it fell upon me, my blood ran cold; and so by degrees—very gradually—I made up my mind to take the life of the old man.
Such reasoning is not indicative of a sane man. Yet the murder did not solve his problem. Even afterward, the narrator is haunted by the man and the imagined beating of his heart. The climax occurs when the narrator can stand the "beating" no more, and he directs the the policemen to tear up the planks to reveal the dismembered corpse--ant the tell-tale heart.
As a reader I would think it is mostly internal, since we obviously think he's mad; he, however, tries to convince us otherwise: "Nervous, very, very dreadfully nervous I had been and am; but why WILL you say that I am mad?"
He then goes on to discuss how it was the old man's eye, which could be considered an external conflict in his mind: "I think it was his eye! Yes, it was this! One of his eyes resembled that of a vulture."
If the conflict is internal, I would argue that no, the internal conflict is not resolved because he's still obviously mad: he is dreadfully nervous when the police arrive, then finally admits to the murder: "It is the beating of his hideous heart!"
If the conflict is external, then I believe the conflict is still not resolved, but more because he is still haunted by the old man even after his death-killing the old man did not stop his external conflict.
Bottom line: seems to me that it is both internal and external, and these conflicts are not resolved when he kills the old man in the climax since he is still mad and is still haunted by him.