Why does he kill the old man whom he "loves?"

The narrator does not love the old man. His claims of love are merely part of the delusional narrative that he gives the reader. Other than the old man's eye, we really don't know what the narrator's motives are. The narrator is completely unreliable so even his claim of the unendurable eye is suspect. Even if the old man had a hideous eye, it certainly is not grounds for murder.