They probably wear a new one for each game unless they insist on an old one...
http://en.allexperts.com/q/Boston-Red-Sox-243/Red-Sox-uniforms.htmI don't have an exact answer for you but I can at least tell you they definitely don't get a new uniform for each game.
The uniforms are made of very high quality and are washed and re-used as often as possible. I would guess that most players only go through a few uniforms a season.
Last week the Red Sox wore green uniforms to honor Red Auerbach of the Celtics. He had recently passed away. Those uniforms will probably only be worn that one time.
Best Wishes,