One probable cause is moisture getting into the wood. This can and most likely is coming from the back side of the wood planks but could also come from the joints and painted side if there are gaps between the wood planks. The oily stuff you have found could be dissolved old paint or tannins from the wood, or a combination of these things. Without being at your home and seeing for myself it is impossible to say for sure.

Fixing it is fairly simple and straightforward.

First any loose paint must be removed. This could be scraping or sanding but either way you most likely have some lead based paint. Familiarize yourself with the problem and take the appropriate steps.

It is a good idea to take a look in the attic space above the ceiling, if there is one, and check for moisture or moisture damage.

The entire ceiling needs priming with a good primer then repainted. With such an old painted ceiling it is possible this problem will return.