CO (Carbon monoxide, not dioxide)
A stove shut down so that oxygen supply is limited, with fuel still burning inside can go through an incomplete burn (oxidation) producing CO rather than CO2. CO is lethally toxic and will kill you to death.
CO will mix and distribute with the air rather than pool high or low.
Keeping a stove in overnight, shut down, is something that I (and others on here) wouldn’t recommend simply for efficiency and sooting reasons: smouldering wood is the most inefficient way of releasing the energy in the fuel. You will only really produce around 20-50% of the potential heat energy from the fuel this way. The most efficient way of extracting heat energy from wood fuel is at a temperature high enough to produce a gasification burn (150-250 degC). LOw temp burning, rather than gasifying volatile compounds in wood, tends to instead produce vapour that condenses in the flue. The condensates are usually quite corrosive as well as have a tendency to solidify and build up increasing the risks of flue fires, flue opening constriction and subsequent reduced draw.
Any which way, a CO alarm is a v good idea in any room with a burning appliance. And is now a building regs requirement for new installs anyway.
EDIT Sorry, too slow typing. T-R will have linked to useful info.