Electrolysis of Water: Overvoltage?
To first order, the power dissipated in an electrochemical cell is equal to that of an electrical circuit: $P=IV$. Thus, as the overpotential necessary to achieve a given current throughput increases, the power consumption increases also.

In the fluid flow analogy, think of an elevated overpotential as a throttling orifice inserted into one of the pipes.  In order to keep the water moving through the pipe at the same volume flow rate, you have to increase the pressure drop across the system.  A greater pressure drop to achieve the same flow rate requires a pump with a greater power rating.