The Courts are often asked to consider copyright claims based on someone having ripped off characters, plot lines and story archs. Recent years have seen JK Rowling and Dan Brown in court on one side or the other.  The general consensus is that there cannot be a non-literal infringement of a story (although this is possible in respect to software). 

In this case, the odds are with CBS due to Axanar's very clear claim to be producing a Star Trek film. What's also exciting copyright lawyers is whether a made-up language such as Klingon can be copyright and if duplicating the made-up Federation insignia will count. My view: the language, possibly and the insignia, definitely.

A wider issue raised by this case, however, is the increasingly high production quality of fan fiction. Expect to see defences raised under the new copyright exemption for parodies, pastiche and satire when fan movies start to gain more YouTube views than the original.