What fails in a centrifugal pump when it runs dry?
Dry Pump failure modes

Some irrigation pumps are created the impeller housing casting an integral part of the motor and depend on the water flow through the pump to dissipate heat instead of using rotor fins and airflow for cooling. The reason for doing this is that they tend to be in high moisture locations or locations where detritus can be sucked into the pump and cause a fire hazard. With no cooling, the field windings will get really hot and can cause the thermal shutdown to trip, causing the pump to cycle on and off as it heats and cools. The heat can destroy the enamel insulation on the coils, causing them to short.

The impeller shaft seal being run dry can burn it out due to friction and lack of water lubrication. You will know when this happens as there will be a leak.

If the pump has a plastic impeller and it gets too hot, it can warp or loosen on the shaft if it's a press-on fit.