Death Note Episode 26

I think it's more to do with him having so much on his mind, then actually not liking her.

As you know shinigami are gods of death and therefore can only use their death notes to kill humans, not save them. L had become suspicious of Misa again, and was going to test the death note and prove the 13 day rule was fake and solidifying the fact that Light & Misa are Kira. L was suppose to catch them and then they would ultimately receive the death penalty for their crimes. By killing L, Rem saved them both from being caught and facing death and gave up her own life by doing so. I honestly don't know if Rem's life span was added to Misa's or not, but yeah that's why Rem died.