Answer
It is unclear whether the narrator would have killed the old man if he did not have an "evil eye." Two pieces of evidence from the text to support this are:
1. The narrator states, "I loved the old man. He had never wronged me. He had never given me insult. For his gold I had no desire. I think it was his eye! yes, it was this!" This suggests that the narrator's motivation for killing the old man was solely based on his obsession with the eye, rather than any personal animosity towards the old man.
2. Later in the story, the narrator says, "If you still think me mad, you will think so no longer when I describe the wise precautions I took for the concealment of the body." This implies that the narrator had already made the decision to kill the old man before he even began to plan how to hide the body.