Home Generator Excess Electricity
The actual alternator in the generator should be harder to turn when there is more current being drawn from it. The current flowing though the coil windings will generate a force trying to turn it.

You can do an experiment with a DC motor. If you short the two leads to the motoro together, you'll see it takes more force to rotate the shaft than if they are not connected.

You can also try turning on the headlights or another large electric load on your car when the engine is idling. You should be able to notice a that the engine is working harder by listening carefully to the sound. 

So, your alternator almost certainly takes more power to turn when you have a larger load. The question really is why that does not show up in higher fuel usage. 

It could just be that the the generator is so inefficient at low power output that the difference in fuel usage due to the increased load is not noticeable. If this is the case, the lost energy is likely leaving the generator as incompletely burned fuel or excess heat in the engine exhaust, rather than anything to do with the electrical part of the generator. 

If you are going to consistently run at low loads, then try to tune the engine to run more efficiently at these small loads. This could be as easy as adjusting the carb depending on what type of engine it is. 

Alternately you could consider getting a smaller generator that runs more efficiently at lower loads. If you use it alot, this could pay for itself in the money you save on fuel in the long run. Honda makes some very nice, very small generators that are impressively efficient and quiet. 

Another option could be to only run your existing generator at full load, but only for short periods of time. You could, for example, turn on the generator for 5 minutes and use all the power output to quickly charge a bank of batteries, then shut off the generator and run your small load from the batteries until they start to go dead, then repeat. This would lower your fuel usage significantly.