Question
how do I get rid of the infestation of gnats in our turtle tank without harming the turtles?

Answer
Are you just finding adults above the water line or do you find larvae in the water?  I've had larvae in the water of my turtle tank.  I do a 100% weekly cleaning and water change which helps.  For the adults, you can buy fly sticky traps to set up near the tank.  Some species of insect larvae are killed by Bt bacteria.  You can add the liquid version of Bt (turtles would eat the mosquito dunks)to the water, and it might kill some species of midge larvae (gnats), caddisfly larvae, etc. It's sold to kill mosquitoes in ponds but kills some other insect larvae as well (and in your tank).

http://www.drsfostersmith.com/product/prod_display.cfm?pcatid=12263&Ntt=microbe%...

If that doesn't work in a week, it's probably not going to.  Just keeping the tank clean is the main thing.  If you put "feeder fish" in with the turtles, the fish may eat some insect larvae too.