Health problems and up keep costs have led to the decision not to use in ground pool. What do I need to do to keep the pool mosquito free while not having to pay the pool service company and the costs associated with running the pump and filter?
You may be able to drain it, but it doesn't take much standing water for mosquitoes to breed so that's an ongoing issue unless you fill it in.
If your don't drain it, you can keep it chlorinated/brominated/ozonated to kill the mosquitoes. That may require running the filter system. If you can keep the pool covered, you can reduce the amount of organic material drifting into the water, which shold make maintaining the chemistry easier. But any standing water on top of the cover risks becoming a skeeter colony again; see above. And you may face Attractive Nuisance legal risks.
You could try turning it into a fishpond, stocking it with enough fish to eat most of the skeeter larvae. Your county extension office, or the equivalent agricultural advisory service in other countries, might be able to tell you whether this would be at all workable. Converting it back into a usable pool might not be easy, though, and the fishpond might still be considered an Attractive Nuisance.
I believe the way most folks solve this is by selling the house to a family who wants the pool and using the proceeds to buy an equivalent or better house, or by selling the land the pool is on (either makes it someone else's problem) or by accepting that it has outlived its usefulness and filling it in.