Keep in mind that many of these generators run at a single speed, to provide max rated power.
Imagine starting your car, and immediately getting on the autobahn while the engine is cold. You can do it, but this can increase the wear and shorten the life of the motor. Same with the generator.
I always let my generators run for a minute or two before applying loads. I also disconnect the loads, then shut the fuel off (on portable generators), which will use the last of the fuel in the carburetor bowl before quitting.
I live in sunny FL, it's usually 80+ degrees when I start the generator, so my warm up time would be shorter than someone living outside of paradise.
I also run my generators at least once a month. This avoids fuel issues.