Well, I'll bite. After almost 40 years in the fire service, I don't think I would recommend it or try it. Been to a lot of chimney fires for a lot of reasons, not cleaned, closed dampers, built on wood floors, cracked flues, animal nests, etc. I don't think your grill would generate enough heat to create a good updraft so you would end up with the smoke from the grill in your house. That would imitate many calls where the damper wasn't opened before lighting the fire. Besides, there is also the issue of carbon monoxide being generated. If you feel your cold chimney generates enough draft to pull the grill exhaust up and out, then you might try it, but only if you have a functioning CO detector close by. The idea sounds very tempting, but I think it would be too dangerous for occupants.