One major clue that the narrator is, in fact, insane is that he decides that he must kill an old man that he "loved" for no other reason than the old man's "vulture" eye. He says, "Whenever it fell upon [him], [his] blood ran cold [. . .]," and this is certainly not a sane response to someone's eye. It's likely that the old man has cataracts, which would account for the eye's paleness and the appearance of a "film over it," and this might be creepy-looking or strange, but it would not be murder-inducing to a sane person.
Further, much of what the narrator says to prove that he is sane actually makes him seem more insane. He talks about "how wisely [he] proceeded" with his plot to kill the old man. He says that he was "never kinder" to the man than he is during the day while, each night, he creeps into the old man's bedroom with a plan to kill him. This attention to detail and "caution," as he calls it, only make him seem more mad. He talks about how slowly he moved his head into the room, insisting, "Would a madman have been so wise as this . . ." This line of argument has the opposite effect from that which he intends.
On the other hand, the narrator talks about his own fears concerning death, and this is a pretty rational fear. He describes the awful moan made by the old man on the fateful night:
I knew it was the groan of mortal terror. It was not a groan of pain or of grief—oh, no!—it was the low stifled sound that arises from the bottom of the soul when overcharged with awe. I knew the sound well. Many a night, just at midnight, when all the world slept, it has welled up from my own bosom, deepening, with its dreadful echo, the terrors that distracted me.
He admits, then, that his fear of death has actually kept him awake at night. Further, this is probably why he needs to kill the old man: the old man reminds the narrator of death because of his age, his disease, and the "vulture" look of his eye (and vultures are associated with death). So, while his fear of death may be a sane one, his response to that fear is insane.