The stove pipe draws smoke out of the stove and your house. The end of the pipe vents the stove outside while bringing in oxygen to the stove. Without venting, toxic carbon monoxide gas can build in your home. This is potentially fatal, and you should have carbon monoxide alarms in your home if you have an oil stove.
When installing stove pipes in your home for oil stoves, you cannot use the same type of pipe you would use for a gas stove or appliance. Gas stove pipes are lighter and not suitable to use with oil stoves. Wood stoves and oil stoves can use the same stove pipe. When these appliances share a flue, the wood stove needs to be connected to the flue lower than the oil stove, according to Carson Dunlap, in "Essentials of Home Inspection: Heating II."
The damper opens and closes to control the size of the opening of the stove pipe. These are used in chimneys to change the amount of air getting to a fire. During operation of an oil stove, you will need to keep the damper fully opened, if you have one, for proper venting of the fire.
The damper is not used during operation of the stove, but you still might want one for your oil stove's ventilation pipe. When you are not using the stove, having a damper to close prevents damage to your stove in the off-season. A stove pipe without a damper could let an animal into your stove that crawled down the stove pipe while you were not using the stove. Heat and cold air from your home can also escape up the stove pipe without a damper. Installing one and keeping it closed will help you to better keep control of your home's temperature.