Originally posted by moonbusHoward Stern does but he is not really a movie star. he is about 1650 in his rating.
At Internet Movie Database under the film The Enforcer (1951), there is a photo of Bogart playing chess back stage between takes.
http://www.imdb.com/media/rm2272368640/tt0043503?ref_=tt_pv_md_1
I believe John Wayne was also a chess player.
Except for Bogart (for whom I wrote something here that he used to be an official of chess federation and had a strength of NM), the most actors are poor chess players or don't even know the rules.
Chess has been connected with espionage since it was "discovered", for example most British and Soviet masters (from NM and specially above that rang) were affiliated with their respective secret services. British masters helped a lot with breaking nazi cipher machine.
There are much more writers and journalists who play chess.