Yes a generator can be run overnight to keep things running while you're asleep as long as the generator is outside and not inside a building which is never should be.
Also check with your local laws and ordinances as they may say whether you can run things such as a generator or other engines that constantly produce noise at night.
Some towns and cities have a noise ordinance however during a natural disaster or emergency they may not enforce those laws on your generator running especially if you require the generator to run for medical equipment.
I've always just shut down my generator before bed when I lived in town and still do here at home in the country because while I'm asleep I do not need the electricity as the refrigerator and freezer will keep cold enough until morning without the need of a generator.
So I would just recommend saving your gasoline or other fuel and avoid the noise and just shut the generator down overnight.
However there's no harm in leaving a generator running overnight.