I usually use a citrus-based stripper to remove paint. If you've tried that already, you'll likely need a liquid solvent to sneak in to the grain.
Knowing what the white paint was could help (enamel? lacquer? water-base latex? oil-base latex?).
The entire spectrum of solvents are out there: denatured (dry) alcohol, mineral spirits, naptha, acetone, lacquer thinner, methyl-ethyl-ketone. All have different absorption rates (penetration), aggressiveness, evaporation rates, etc. I have had some success using three parts lacquer thinner with one part dry alcohol. The lacquer thinner is decently aggressive and the alcohol helps it penetrate. Many other combinations are useful.
Always use this stuff in a thoroughly ventillated area. Use it only long enough to do your cleaning since, depending on the species of wood, these organic solvents can pull more sap out than you might desire. Light rubbing with a 3M abrasive pad can help and always be sure to protect your hands from the solvent. I'll save my kidney failure story for another time.
When the wood cleans up (I'm an optimist!) permit a day or five to let the solvent evaporate before application of your finish oil.
Good luck!