You shouldn’t ever use a charcoal burner indoors, as it produces a lot of carbon monoxide.1 This is an odorless, colorless gas that can be fatal in large doses. Carbon monoxide poisoning can make you dizzy and confused, and you may pass out. It’s particularly dangerous if you’re intoxicated or asleep, as you could inhale fatal quantities of carbon monoxide without knowing.2
You can use a propane camping grill indoors, but you should ensure you’re cooking in a well-ventilated area and that you’re monitoring the grill carefully.