Should I use a GoPro Hero 3 Black for recording drama?
The GoPro is really more designed as a small, rugged, action camera.  It really isn't designed for recording a theater production and you would be able to get much better quality out of any number of camcorders in that price range.  You really want something with an optical zoom on it.

Either something like a Canon T3i or T2i would probably handle what you need.  Alternately, Panasonic makes some pretty good video cameras in that price range.

It is also worth noting that the GoPro has no screen so you would need to buy a separate screen to know what it is that you are actually shooting so that you could aim it.  There is one available from GoPro, but it adds around $80 to the cost.