Any type of oil application will at first be sucked
into the slug (clay composition) to only later come
to the surface when moisture is present. This, depending on the pigment
composition, could move that pigment within and on
the surface of the chip causing what may seem to be
a stain. Oil is oil, and it's lighter than water
along with being a solvent in some cases. Considering what you are housing the chip in is also a very important factor if oil was applied to the chip. That material can break-down and damage the chip, too.

As far as what type of soap to use,
which would you rather use on your face: dishwashing
which is designed to cut dried-on cooking grease, or
a mild face soap? Besides, if what you want removed
cannot be done to a dirty chip with just a simple
toothbrush - leave it. When a chip is excessivly
dirty it will probably be somewhat worn, and a
clean, worn chip doesn't really look too cool.
There are many other factors concerning cleaning,
such as disrupting the glue under the inlay causing
the inlay to pop once the chip is dried and a ton of
other stuff. Of course very few people are willing
to tell you what went wrong when they tried to clean
a chip that was worth a few bucks.
JB