Does a vegetable oil lamp produce carbon monoxide?

Depends if the oil is a saturated fat or an unsaturated fat. Unsaturated fats have a higher carbon content so they sometimes produce carbon monoxide but rarely.

Vegetables oils frequently contain unsaturated fats.

Anything that burns CAN produce Carbon Monoxide but if it has an adequate flow of air, it is much less likely to happen. If you burn this oil lamp in a small, badly ventilated room, you will almost certainly produce CO in time, but if you use it in an open field you wouldn't be likely to, and, even if you did, you would be unlikely to be effected!

Check how "air tight" the room is, and, if it has a good flow of air, give it a go! Remember that these lamps were never designed to be used in a modern hermetically sealed home, but in cold rooms with the wind howling through the cracks! At the first sign of a headache or drowsiness, open the windows and doors, and let some air in!