Avoid putting any sort of oil on your bird. This will spread very rapidly like when you drip a drop of gasoline on your car. Petroleum likes to be in a single layer. (hence why ocean spills get so large) This causes the bird to lose insulation properties of their feathers and they will freeze to death. If it happens use warm water and Dawn dish soap or a mild dish soap to remove it and help dry the bird with a hair dryer on low and far enough away so the bird won’t get burned. Birds usually LOVE the blow dryer but be careful. Birds are sensitive to temperature changes.
Just like any accident, its never planned and always happens at the worst possible time. Like Christmas. Many times in these instances you have to think on your feet. Like the time when our new baby bird, Herbie, fell… in gravy.