Electrical Load increase on Generator = More Fuel Consumption
The generator has a speed governor. 

As electrical load increases the mechanical load increases. If the throttle position were fixed the speed would drop. The governor opens the throttle to allow more fuel in to maintain speed at setpoint.



Figure 1. Mechanical speed governor. Source.

Figure 1 shows an old-fashioned mechanical governor.


As the shaft rotation speed increases the flyweights are thrown outwards, lift the control sleeve against the speeder spring and drive the crank to shut off the fuel valve.
If the shaft slows down the speeder spring pushes the control sleeve back down and the crank opens the fuel rack again.
Spring pressure is controlled by the speed control lever to set the running speed.


Modern speed regulators are electronic. They monitor the output frequency and adjust the fuel valve electrically.