Can I dry items that have been exposed to cooking oil?

No, do not dry anything that has ever had any type of oil on it, including cooking oils. No washer can completely remove oil, and oil-affected items can ignite spontaneously when exposed to the heat in a tumble dryer. If it is unavoidable to dry such items, wash them in hot water with extra detergent before drying, though this reduces but does not eliminate the hazard.