The best thing I have found for removing a fat or oil stain -- or the oily component of a more complex stain such as salad dressing or gravy (once it's dry) -- is to rub cornstarch (or cornstarch powder, or even talcum powder) into the stain, let it sit for at least 20 minutes, and then brush it off with a dry washcloth or soft dry brush. You may need to repeat the treatment several times to fully remove the fat. It works especially well on silk and wool. So well, in fact, that further washing (or dry cleaning) is not needed. You can even speed up the process by placing a paper towel above and below the treated spot, and going over the area with a hot steam iron.