Hi, Coy. First, don't sand. You need to strip the oil off until you get down to the wood but I don't believe in sanding the wood unless necessary for cosmetic reasons.
Take the piece outside and wash the surface well with one of the organic solvents like naptha. Don't forget to stay away from sources of ignition and to leave the rags outside for a couple of days to dry. I noticed a Park's product yesterday when walking through Lowe's that is supposed to be made especially for this also. I haven't tried it but it sounds interesting.
Then, put of some Citristrip Paint Remover Wash in a small area. I have some doubts that this will be enough alone, but you might get lucky. However, the good thing is that this will make the white stand out. If this doesn't work, try a brass brush (gently) with the Paint Remover Wash and brush with the grain. It would help to know what kind of wood it was, but it is probably oak and if you are careful, you should be able to get it out. However if it still doesn't, you will have to use a stripper but this should take a half hour at most. Then go over it with the Paint Remover Wash to get out any remainder and again the Wash will make the white stand out.
If you went too strong with the brass brush, put on some Bartley Paste Wood Filler and after it has set up, remove the excess with a piece of burlap. You may have to lightly stain depending on the color of the wood and how much of the Bartley's is in the grain. Don't use any other product than the Bartley, which is available from Woodcraft (www.woodcraft.com).