Yes, it does. Logically, as you require more power, the power has to come from fuel. There is no other source. An old-school genset runs at constant RPM to get the frequency of the alternating current right. This is usually 3600 RPM with small, gasoline-powered generators and gaseous fuel home stand-by generators. Spending more money can get a higher quality generator that may run at half that speed. There is significant fuel burned with no load at all. Adding electrical load will then add to the fuel consumption in a fairly proportional way. Lots of manufacturers will give the fuel consumption of their equipment at half and full load. This relationship should be pretty similar to any other internal combustion engine-fueled genset.

Newer, inverter gensets spin at lower rpm at no and low load so under these conditions they burn less fuel than their old-school friends. (They are much quieter too!) At some point as the load increases, the difference in fuel consumption, however, will become very small.

One thing that this means is that running the generator sporadically, if that is possible, will save fuel as well as your ears. For example, if you need to keep your freezer frozen and you run a well pump, you may run the freezer temperature down lower than usual in the evening and shut the genset off for a few hours. Nobody will be needing running water and the freezer will hold cold enough.

Nice post by ionized_gw above.

Spot on!

thank you very much, excellent response. We just went through hurricane IRMA and had no power for 4 days. I have a Generac GP5500. It ran pretty much constantly for that four days. It has a 7gal tank I was getting right at 9 hrs per tank if filled right to brim.