Facebook did take the that page down but our other account remains active.
It wouldn't surprise me if CBS (or even Paramount, who are busy promoting their movie) did ask Facebook to shut it down. They probably wouldn't have to send a C&D to do it either. Case in point: for 4 years, I helped run a Facebook fan page for the 1987 "Masters of the Universe" movie. We posted tons of stuff from the movie, from pictures to interviews, and never got any flack for it. In October of 2012, the same week that the MOTU Blu-ray was relased, our fanpage got shut down. No reason was ever given to us, but Warner Brothers did put up a MOTU 87 Facebook page to promote their movie. Our guess was that someone at Warners asked for our fan page to be taken down, so that they wouldn't have a fan page competing with their $$$ page $$$. (which was kind of silly, cause we would have promoted the hell out of the Blu-ray for free).
Fan films that have "Star Trek" in the title and have Facebook pages could be seen as competiton to people who work on 'official' Star Trek, in the sense that it could divert peoples attention to what it is that the studio is trying to promote. If they are trying to promote a multi-million dollar movie, they proably don't want people surfing through various Star Trek groups to find their film page.