During the 1960s, a now-defunct organization of educators in theatre addressed this very question. After discussion of theatre, theater, public perception and educational values, they arrived at an agreement. The guideline that they promulgated, and that many theatre planners and consultants generally follow, is this:

Theatre: refers in general to the art of live performance. Also, the building in which live performance takes place.

Theater: refers to the building in which movies are displayed.

In usage, then, you would go to the theater to watch the film version of Hamlet and to the theatre to see it performed on stage.

Theatre (or Theater) History

Some history of the word, from the The American Heritage® Dictionary of the English Language, Fourth Edition, Copyright © 2000 by Houghton Mifflin Company, as published on Dictionary.com:

[Middle English theatre, from Old French, from Latin thetrum, from Greek thetron, from thesthai, to watch, from the, a viewing.]

Word History: Theories about the development of the theater in the West generally begin with Greek drama; this is etymologically appropriate as well as historically correct, since the words theory and theater are related through their Greek sources. The Greek ancestor of theater is thetron, “a place for seeing, especially for dramatic representation, theater.” Thetron is derived from the verb thesthai, “to gaze at, contemplate, view as spectators, especially in the theater,” from the, “a viewing.” The Greek ancestor of theory is theri, which meant among other things “the sending of theroi (state ambassadors sent to consult oracles or attend games),” “the act of being a spectator at the theater or games,” “viewing,” “contemplation by the mind,” and “theory or speculation.” The source of theri is theros, “an envoy sent to consult an oracle, spectator,” a compound of the, “viewing,” and -oros, “seeing.” It is thus fitting to elaborate theories about culture while seeing a play in a theater.