Theoretically yes, in practice no. There is (according to my sensors) a fairly large CO2 increase inside a room when a modern furnace (with external exhaust) is running. I've confirmed this with several units (all made in the last 10 years), and it's not that the windows are closed - when the furnace turns off, the CO2 drops. And it's not that the exhaust is placed in a bad spot either.

Could also just be the furnace. Incomplete combustion conditions can give rise to the symptoms mentioned here.

I doubt it's the furnace, unless there's a serial defect/recall.

The problem is likely "between the keyboard and the chair." ;-)

I had a gas furnace that wasn't properly maintained as far as cleaning. Result: insufficient air flow for full combustion. Secondary result: CO build up in basement space. Tertiary result: asthma-like symptoms for me.

I believe it.

Yes, fossil fuels are the best to keep pollution away, just need to installed perfectly, configured and maintained regularly, monitored to make sure everything is running correctly, and have additional properties lying around vacant just in case there are leaks, misconfigurations, poor installation, etc. But we must use fossil fuels, there are no other options!

Your control for this test should be (and maybe was, you don't say) running the furnace circulation fan without running the burner. CO2 levels are unlikely to be uniform throughout a building, and thus mixing will change (raise, lower) the CO2 levels depending on where you're measuring.