So, you know how there's a 'fire triangle' for combustion? Oxidizer, fuel, heat (and ignition)?

I'd say there's another one for complete combustion: even mixing of fuel and oxidizer, sufficient but not overabundance of ambient airflow to equilibrate volume change by exhaust gases from combustion reaction, and sufficiently high total concentration of oxidizer and fuel in flame environment to maintain combustion with the energy released by the reaction.

In a normal fireplace, or a gas furnace, none of these conditions are met perfectly throughout the region where combustion is happening. (It's quite good in modern, to-code gas appliances compared to a wood fire or a really sooty, misadjusted propane burner, however).

For example, that means there are regions where there are oxidizer and fuel (usually the gaseous reactants) are well mixed, but too dilute with ambient air to burn. Other regions might have too much fuel and not enough oxidizer because of turbulent mixing with the ambient air, but plenty of heat, so the fuel will undergo reducing reactions instead of oxidizing, which turns hydrocarbons into hydrogen (which will burn somewhere in the fire) and carbon (which is soot), and then that carbon is able to cool before it's mixed with enough oxygen to combust.

Imagine trying to mix two colors of cold wax in a pot on the stove just by heating it up and letting the bubbling and boiling do the work. That's about the best a wood fire can hope for in terms of even mixing of the flammable gases coming off of logs and coals with the air it needs to combust. There are all sorts of pockets of reducing and oxidizing environments in there, and as a result you get soot as well as weird, icky nitrogen compounds which give us smog etc.