She initially only went out with him because she felt sorry for Moe that he couldn't get anyone. In spite of that beginning, though, they were a happy couple for a long time. Renee broke up with Moe after she found out that Moe wouldn't bail Homer out of jail, instead choosing to spend money on a trip to Hawaii for the two of them. Renee was shocked by Moe's callous treatment of his friend. When Moe started to tell her about a plan he had (involving corpses and burning down the bar) to both bail out Homer and take the trip, she became disgusted, asked for her keys, broke up with Moe and left.