My oil furnace may have become wet from a dripping pipe overhead. I am pretty sure the burner got wet. When I fired up the burner after the dripping was stopped, white smoke with an oily smell entered the flue pipe. Is this an indication that the burner needs to dry out more?

This sounds like a issue with the burner not lighting off. When there is no flame in a oil burner and the unit is running it will produce white smoke and smell like oil especially if the combustion chamber is hot.

I would call for ON SITE professional Service before something bad happens. If they light off in this condition it can blow the flue pipe out of the chimney, back flash into the combustion chamber and blow open all burner access doors unless they are tightened down.

Calling for service will also let the service person check to make sure there is no other damage to the unit from the water.