That would be extremely bad. Mosquitoes are big pollinators, and are a major food source for frogs, dragonflies, and other critters. Mosquitoes disappearing would just cause a domino effect. We already in a biodiversity crisis, this idea of eradicating animals needs to stop. Apparently people still refuse to learn this but, whatever we do to the ecosystem will bite us in the ass.
Not to mention that the vast majority of mosquitoes do not even bite people. And only a small handful of species that do are really problematic, such as Aedes aegypti. Your statement would make a bit more sense if you suggested to eradicate the disease-carrying mosquitoes. Problem is, once we leave that niche open there is a chance that something even worse would take their place.
Even then, it would be far easier to come up with better ways of repelling mosquitoes or prevent the spread of diseases such as malaria than to attempt to kill them all.
Mosquitoes should only be culled in places where they are overpopulated and/or invasive (again, A. aegypti is an example in the U.S).