The old man is even more of a mystery than the narrator, partly because we only see him through the narrator's skewed perspective. We know he has money (the narrator shows the old man's "treasures" to the police). We also know he has a blue eye that the narrator is afraid of, and which fits the description of a corneal ulcer. We know he's old, and that he's a fairly sound sleeper.
Not much meat for a character study, right? Luckily, we're given a few more hints to work from.
According to the narrator, the old man suspects nothing because the narrator was super duper nice to him the week before he killed him. We can't prove the old man wasn't suspicious, but because he leaves his bedroom door unlocked we can assume it. We know the man isn't naturally trusting – he's afraid of robbers.
But, it seems he does trust the narrator enough to give him the run of the house while he sleeps:
To think that there I was, opening the door, little by little, and he not even to dream of my secret deeds or thoughts. (4)
Unless the old man is a poor judge of character, or senile, his trust suggests that the narrator really is capable of acting sanely...and that the old man feels warmly towards him.
Nothing the narrator tells us about the old man fits our idea of "madness" or "insanity"—the insane quotient in this story is all pretty much used up by the narrator himself—but the old man does fit neatly into the narrator's definition of madness. He has "destroyed" or "dulled" senses and, to quote our narrator himself,
Madmen know nothing. (2)
The old man only hears the narrator on the eighth night, he doesn't seem to have the slightest idea what's going on around him, and he's incapable of defending himself. Perhaps the narrator is slyly hinting that he thinks the old man is "mad." This makes us wonder if the old man was senile or dependent on the narrator's care.
If so, this adds a new dimension to the creepiness afoot in "Tell-Tale Heart," and paints the narrator in an even more negative light.
We know that at least one neighbor is suspicious of the goings on in the house of the old man and the narrator. Otherwise, he or she would not have been so quick to call the cops after hearing a little scream, and wouldn't have been able to convince the powers that be to send not one or two, but three policemen.
We don't know if this suspicion is directed toward the old man or toward the narrator—or both. But it's possible that the narrator wasn't the only one afraid of the old man's eye. The old man could be an alienated figure both in and out of the home, and thus the narrator's murder of him could be symbolic of prejudices and abuses that stem from physical "difference."