How do I manipulate a large number of moving platforms?
In terms of cost, perhaps it would be best to build a 4x4 prototype.

My first thought was to use inexpensive servos with a flexible rod inside a sheath, perhaps a wire, like a bicycle brake/gear.  The amount of movement could then be governed by the size of the armature on an inexpensive servo.  

Servos will position themselves after being given the command, so there's no need to wait and do them one-by-one.  Fire off the command, then use some sort of multiplexed output like multiple 74HC595 shift registers to connect to the next one.

If the tiles were spring loaded, such that they returned to the topmost position by themselves, only a simple pull-string is needed to adjust the tile height.  Even a very light filament (like fishing line) would be strong enough for the pull.  This would allow the servos to be positioned even some metres away from the table.