How do I remove (food) oil stain from a wood table?
I think there's something in between getting the stain out and "refinishing" the table, and that's applying a food-safe oil finish to the entire table.  You could wipe on (and back off) some linseed oil, tung oil, or even more salad dressing for that matter.  This would be a lot less effort than any kind of refinishing and should do a lot to make the stains look less obvious.

Note that I'm assuming that, since the wood table has no current finish it's not a museum piece or anything and is meant to look a little rustic.