Given The Same Load, Will A Larger Inverter Generator Burn More Or Less Fuel?
There is a sweet spot in between overloading and underloading the generator (running it a bit higher or lower than its optimum operating point).

It means that if you overload the small generator, or underload the large generator then your fuel efficiency will not be as good as it could be. Which one is worse is up in the air because it becomes a matter of degree as to whether the small generator is more overloaded than the large generator is underloaded.

A larger generator certainly will burn more fuel than a small one if you're just idling. So if you are idling more often than not, perhaps it is better to pick a smaller generator that gets is loaded a bit higher than its operating point for maximum fuel efficiency. On the other hand, if your generator is heavily loaded most of the time, this is probably not the best solution and you instead want to pick something that is nearer its maximum fuel efficiency when loaded.

In your example of a 1000W load, I would lean toward it being more fuel efficient on the 3000W generator just because 3000W is already so much larger than the required 1000W, and 4000W is even larger so you're spinning around a lot of dead weight.

The rated wattage on the generator is probably the point of maximum continuous power output, not the point of maximum fuel efficiency. In the unlikely event it is the operating point of maximum fuel efficiency, then 3000W will obviously be better since 3000W is closer to 1000W than 4000W is.

But in the more likely event that the rated wattage is the maximum output power, I still lean towards 3000W because I find it hard to believe that any machine's optimum fuel efficiency occurs at something as low as 30% rated load, let alone 25%. It could be, but my gut feeling is that it doesn't. Someone who knows more about generators and engines will have to chime in.