This is a fascinating question to consider when we think about this excellent example of Gothic short fiction. Somehow, the normal categories and definitions of protagonist and antagonist don't seem to really fit this tale, and so we are forced to be rather inventive in considering conflict and opposition. It would be tremendously easy to say that the nameless old man was the antagonist, however we would need to consider in what sense he is opposed to the protagonist. Certainly, the narrator himself does not recognise any source of conflict between them:
I loved the old man. He had never wronged me. He had never given me insult. For his gold I had no desire.
What it is that seems to drive the murderous intentions of the narrator is the morbid and strange repulsion he has with the old man's 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, and thus rid myself of the eye forever.
This bizarre hatred of the eye combined with the beating of the "tell-tale heart" suggests to me that actually, the conflict that is present in the story is some kind of internal conflict that pits the sanity and the insanity of the narrator against each other. You might want to consider the increasingly frenzied response of the narrator at the end of the short story as he sits listening to the heart and the police refuse to go:
I foamed--I raved--I swore! I swung upon the chair upon which I had been sitting, and grated it upon the boards, but the noise arose over all and continually increased.
He is the only one to hear this increasing din, suggesting that perhaps this sound and the narrator's madness is the result of some form of conscience expressing itself, emerging from the narrator's psyche.