There is a way of making the statues move, but it would make the game too long. It's using MCEdit. You choose a statue in game, close the game and in MCEdit you move it, then come back to the game and make this until the game ends.

It would be possible to make the pieces move with either World edit's copy and paste option or the Ugocraft mod.

wow

ok

I bet you could actually make them move using command blocks.
You could have buttons with stuff written like "Knight to E-5" which would activate a command block that tps you to an area with Knight already at E-5, though you would have to make 100's, maybe 1000's, of different chess sets with each piece in a different position, but it isnt impossible. :D