Wood Heaters, page-135

Ashes, there is no way you will ever suffocate yourself with a slow combustion heater, they heat via convection from the firebox so carbon monoxide is not a possibility and the air they draw in to burn is minimal compared with an open fire. It maybe possible to run low on oxygen if you had one in a sealed shipping container or an aircraft hull but in an ordinary house if it was so well sealed that no air could enter then the walls would cave in before you would suffocate.