I'm currently researching for an article on theater trailers for "American Theater." And somehow, in my research, it's turned into an exercise in how to properly film theater. I won't tell you what my findings are until the article runs in the magazine (at some point). But I will say this: close-ups are key. Some plays lend itself better to being filmed than others. For a momentous piece (say "Sunday in the Park with George"), rendering the scale of the set is slightly problematic because the size of the stage and set is minimized on the television screen, everything looks smaller.
But for smaller plays, it seems that a good close-up can make the play more effective. Case in point, a play by the Civilians (a New York-based documentary theater company) called "You Better Sit Down: Tales From My Parents' Divorce."
It's four actors, recounting the stories of how their parents met, fell in love, divorced and what they fought over. The actors play their parents (very convincingly). Seeing the work in the Flea Theatre's small black-box space in Soho, the nuances and humor didn't quite hit me as I was sitting towards the back. But seeing the filmed footage of the play, from the comfort of my couch...that was something different. And for me, the piece was funnier and more biting, and it seemed they were speaking to me.
Or it may be because when I saw the play, it was towards the end of a long workday and humor doesn't always hit you when you're tired. But take a look for yourself.