I've seen similar on some other late model cars. It seems the coolant flow is such that it ALWAYS flows through the heat exchanger, even with the temp set all the way down. I don't know if this is true of the Stingray, but it's likely. The heater box should have a door of some sort that opens / closes to vary the amount of heated air that mixes with the cool(er) outside air. If that door is not closing properly, it could cause the problem you're having.
OTOH, it may be that the system never completely shuts off the flow of heated air, by design. Since most people run the air conditioning all the time, Chevy could have decided it didn't matter. That seems unlikely, though, as it would make the A/C run more, which would use more fuel. But you could bypass the heater core during the summer by disconnect the hoses that attach to it and plugging a short piece of pipe between them. But you'll have no heat on those occasional chilly mornings, and defrost might not be very effective on really humid days.
On the Miata forum, someone mounted a valve and a couple of fittings such that with the valve in one position, water flowed through the heater core, and with it in the other it bypassed the heater and just flowed backed into the cooling system. Something similar should be possible on the Stingray, if the hoses are relatively accessible. Use an electrically activated valve, and you could even control it from inside the car.
Here's the thread from the Miata forum. Start at post # 57.
http://forum.miata.net/vb/showthread.php?t=375731&page=3
I'd start by having the dealer make sure the heater airflow door is working properly.