The world could very likely get by just fine without having mosquitoes in it although mosquitoes help feed fish, birds etc.
But those animals and fish could likely feed on other food if we did eliminate mosquitoes.
The mosquitoes are basically a fly that is in the family Culicidae, and there are over 3500 species of the mosquitoes.
It would be impossible to actually eliminate every single mosquito so sadly we won't be able to make mosquitoes go completely extinct at least anytime soon.
Maybe in a million years or more the mosquitoes may fully go away but highly doubtful and by then it won't matter much to me anyway because I'll be dead and then the bugs will be feeding on me underground.
But according to science the mosquitoes do play a very important role in the ecosystem and help provide the fish and birds with nutrition.
Mosquitoes also pollinate some plants but then again so do bees so even if mosquitoes were no longer around the bees would still pollinate the plants as long as bees were still around.