As everybody knows, Eartha Kitt played Catwoman in the third and ultimately final season. It has long been said that because her filming schedule for McKenna's Gold, it conflicted with the schedule for the series. However, this has been called into question by an analysis (http://www.66batman.com/forums/viewtopic.php?t=860) of the evidence. Notably, the fact that Burgess Meredith had no problem appearing in both McKenna's Gold and Season 3 of Batman).

On theory is that Julie Newmar didn't do Season 3 because she would've inevitably toward over Yvonne Craig. According to IMDb, Newmar is 5'11", but Yvonne Craig was only 5'4". Conversely, Eartha Kitt was a mere 5'2". Basically, next to Julie, Yvonne would have looked more like Catwoman's protégé instead of a a heroic peer that was supposed to stop Catwoman's villainous ways.

Another reason why Julie might not have been asked back is due to the sexual chemistry between her character and Batman. In the third season, the producers were often pushing the relationship between Bruce and Barbara, so they might not have wanted Julie there to complicate (https://forums.stevehoffman.tv/threads/tv-stars-who-were-fired.589666/page-9#post-16166113) that dynamic.