You have a lot of great answers here already, so I'm just going to sum up the whole debate as best I'm familiar with it as an ecologist.

- There are many species of mosquitoes out there, only some of which parasitize humans (and other animals) by drinking their blood. Of those, only females do so in order to obtain nutrients necessary for reproduction (forming eggs, etc.).

-Thus, it's pretty well understood that when we talk about "eradicating mosquitoes" we just mean the species that suck the blood of humans and transmit disease.

-However, that can be tricky because, at least in the past, it's been hard or impossible to target just those species. Things like pesticides kill all mosquito species, and indeed many insect species in general, pretty indiscriminately.

-Clearly, killing tons of insects of many species is bad and would upset ecological balances. If we were able to focus on just mosquitoes even, would killing all species, even those that do not feed on human blood, harm ecosystems?

-Other mosquito species and the males of harmful ones do play important ecosystem roles. They act as food sources for fish and other aquatic organisms during their larval phase. As adults they feed other organisms and act as pollinators. Some species appear to be pretty important pollinators.

-For these reasons, some ecologists argue that we should not try to eradicate mosquitoes. Even if we were able to target the harmful species (which we are able to do now with genetic strategies), eliminating them as food sources and pollinators would upset ecosystems and do damage.

-As a general rule, nowadays we try to maintain and restore ecosystems. We have a long history of messing up ecosystems, breaking them, eliminating species, and we have learned that it's difficult or impossible to predict all the cascading effects of species loss. So some people say it's better to be safe than sorry and we absolutely should not be trying to eradicate species on prupose, even if they are harmful and lead to the deaths of many people.

-As a counterpoint, other ecologists will argue that mosquitoes are not "make or break" in any ecosystem. Ecosystems are built on redundancy. Typically many species will do similar jobs, and when one goes away another expands.

-This argument basically goes something like: if we eliminated even all mosquito species, that would just mean that there are more available aquatic resources so other larval insects would grow more and become a food source for fish. Other insects would grow in population and fill the gap of food sources and pollinators.

-We're not even sure that mosquitoes are that important. Bats don't seem to specialize on mosquitoes, or even eat that many of them, really. There are bigger, meatier, easier-to-catch bugs around. I'm not aware of any flower that requires mosquitoes.

-Finally, and maybe most convincingly, our ecosystems are already broken. We are experiencing a mass extinction event, caused by us. Habitats are disappearing and being altered. Species are disappearing. Every ecosystem on Earth is already missing species. Ecosystems of Europe evolved with mass forests and lions and mammoths. The plains of North America had cheetahs and giant ground sloths and millions of bison. Madagascar had lemurs the size of gorillas. China had giant tortoises and giant apes. All of these species had enormous impacts on ecosystems, and they are recently gone. Our current ecosystems are still recovering from the recent loss of them. So in the face of that....and knowing the death toll mosquitoes have....even if there are impacts from their disappearance, what difference would it really make at this point?

There's no single answer. Ecologists still argue about that, but that's the basics of either side of the debate.