If you cool the exhaust gases too much they will condense in the flu, and the condensation is incredibly corrosive. Stoves/furnaces etc. (except for condensing ones) intentionally waste a lot of heat intentionally keeping the exhaust gases hot just to prevent this.
The hot exhaust also helps create airflow that moves the combustion products out of your living space.
It would be neat to recover otherwise totally wasted heat, just be careful.
