An interesting question.
Back then - probably not. Nowadays - Yes some do but really, it's only the big boys that do it. For instance - the RSC and their production of Hamlet with Davd Tennant. (Although my understanding is that the filmed version came about well after the stage production finished).
Both the RSC and the National Theatre have picked one night where a production will be transmitted live to cinemas and, given that DVDs are on sale in their shops I would imagine that there is an element of filimg at the same time (if only to allow for a slight delay in transmission in case of glitches). They only pick certain productions though - which generally are the ones with the big names starring. I don't imagine that many smaller companies would have the facilities or the funds to do anything similar.