[Zanthis]

A quick test in the simulator reveals you do in fact lose supply for firing weapons. You can fire even at zero supply. However, as soon as you drop to zero supply, your maximum movement in combat drops to 1. It fails to display this properly in the upper right display window until after you move. It correctly displays this if you right-click on the ship.

Note that while your maximum movement is reduced to one, you still have any unused movement to spend for that turn. So, you might be at 4/1 movement upon hitting zero supply.

There is no effect on shields in combat.