Unfortunately no, it's a sealed system. The combustion air does not mix with the heating/ cooling air. But I like the way you think.

I have built and installed these in Providence. The building it is next to has a gas fired boiler or other heating equipment in the sub basement. These boilers require both intake air for the gas combustion and an exhaust. The exhaust is hot and rises. It comes out the very top where the vent is narrower. The intake air goes in lower where the vent widens. It is essentially an exhaust pipe running in the middle of a larger intake pipe.