Flippy and Mime, while both friendly towards other characters, are shown to dislike each other. Most of the conflict is because of Mime making too much noise, annoying Flippy. However friendship can (albeit rarely) occur between the two.
Flippy clearly is not fond of Mime, whether or not he is flipped out. In Keepin' it Reel, he is obviously annoyed when Mime is eating his imaginary popcorn too loudly. Flippy also gets very annoyed with Mime throughout Random Acts of Silence, where Mime continually makes loud noises when at a library, at which the former is a librarian. In Party Animal, he is also indifferent to the fact that he had killed Mime while flipped out, even drinking his remains after using one of his antlers to stir it.
On a more positive note, Flippy seems impressed with Mime's skills in Double Whammy Part I, and seems horrified that his evil side had killed him.
On the contrary, Mime does not seem to harbor many negative feelings towards Flippy, unlike the latter. He seems pleased when Flippy sharpens his pencil in Random Acts of Silence. He also helps out at his party in Party Animal, but he doesn't acknowledge Flippy at his own birthday party and isn't around to sing to him (Even if he can't speak). However, he does not seem to heed Flippy's request to be quiet in either Random Acts of Silence or Keepin' it Reel, showing disrespect towards Flippy.