Delivery pumps (pumping water to your brush) are diaphragm type. There is a plate inside the head of the pump which moves up and down. In the plate (the 'diaphragm') are 'flap' valves which are operated by the water travelling through them. If you run the pump dry all that happens is the diaphragm bobs up and down and the valves sit in the diaphragm and don't move.
Other than an outside chance of it overheating, there's nothing to go wrong so it shouldn't do it any harm to run dry.
Rotary and centrifugal types sometimes rely on water to lubricate the spindles, but generally speaking diaphragm pumps don't need lubrication.