For what I know, the most poisonous place of your house is the kitchen. Even with a perfectly regulated methane burning stove, the CO2 and other combustion gases that you breathe are not good for you, even if they seem "odourless". Even if you have an electric stove, the vapour of oils when you fry and cook are not good for you.
Ventilation is necessary whenever you cook. Cooking anything in a basement without ventilation, or in a kitchen with windows closed and no forced ventilation, is not healthy. Even in cold months, it is good to leave the window a bit open and the fume extractor on.
All that said, using a "turkey frier", or any flame without ventilation is not a good idea, not just for the CO or the CO2, but because combustion is never perfect and you are always bound to breath some combustion pollutant, not just CO2 and water.
When I use my gas oven to sanitize bottles, I always feel and recognise the combustion gases (the stove is new) and I let the window a bit open. Some people detect the "smokes" less than other, but they breathe them just like anybody else.
I learn from this thread that there is people using "turkey fryers" indoors or in a basement. So far I gave for granted that this is considered an option available only to people having a patio or working in the garage (with the entrance open) or in a shed with good ventilation.