Removing the cat’s eye is the beginning of the end, so to speak, for the narrator. Until that moment, the cat had thus far escaped his drunken cruelty, so this act signifies that the narrator had reached a certain mental point: there were no longer any limits to his violent behavior. After cutting out the cat’s eye, it starts to avoid him any time he is around, and it is that reaction that angers the narrator enough to hang the cat from the tree outside.
Although the narrator states that he is drawing no causation between what he did to Pluto and what happened to him after, the readers are led to believe that the cat that follows the narrator home from the bar is a form of revenge for what happened to Pluto. After all is said and done, the new cat is why the narrator murders his wife (he does so in rage after she stops him from killing the cat), and then it is the cat’s meowing that alerts the police to where the narrator hid the body.
So the removal of the cat’s eye is so important because, after all is said and done, it leads to a series of events that end with the narrator’s own execution.