Have you any suggestion for restoring the panels to their original condition? If not, can you suggest a place where I might find similar panels for replacement?
The wood has been penetrated all the way through, so that even though only the surface is stained, the petroleum-based product will remain inside the wood fibers. Another leak will cause the substance to be drawn out to the outside of the wood, making the stain reappear. So you really have three problems to solve. One, stop the source of the leaking; two, remove the stain; and three, seal the wood sufficiently with a penetrating sealer that "crowds out" the fibers that were cleaned, disallowing the staining tar to move to the wood surface again.
Apply mineral spirits to a clean, dry cloth and gently rub the surface. You may need to repeat this several times with additional clean cloth rags, since you must avoid swishing the stain around and spreading it either into more surface fibers, or making the stain larger. Once you have gotten as much as you can out of the wood, allow the area to completely dry. Do this in a well-ventilated room to avoid inhaling or igniting the dangerous fumes.
Once the area is dry, gently sand with fine sandpaper and apply a seal coat of linseed oil, if the whole ceiling has never been stained or sealed. If the ceiling has been finished before, try to match the stain or you will have to sand and refinish the entire ceiling. When you are satisfied with the stain's removal, coat the dry ceiling with polyurethane in at least one coat.
Once you do this you can stare at your ceiling often and know things are looking up. Good luck!