It is not safe to leave a wood stove burning at night. The main reason is that it's easy to forget that you have a fire going, and the next thing you know, the house is on fire. Another reason it's not safe to leave a wood stove burning at night is because of the risk of carbon monoxide poisoning. Carbon monoxide is an odorless, colorless gas that can be deadly if inhaled in high concentrations. So even if you do remember to put out the fire before bed, there's still a chance of getting poisoned by carbon monoxide.
It is safe to leave a wood stove burning at night as long as you follow some basic safety precautions. Make sure the chimney is clean and in good condition so that the smoke can escape properly, and always place the stove on a sturdy surface that won't catch on fire. Be sure to have a fire extinguisher nearby in case of an emergency.