I've read in several places that oil should not be used on Japanese waterstones, but none of the sources explain the reasoning behind this. What, if any, damage would using oil cause and would it be reparable?
Japanese style whetstones made of natural material, also known as "waterstones", rely on water to make a paste from the material, and this paste is abrasive and aids in sharpening. Oil is not as effective as water in making this "mud" as the stone material isn't as soluble in oil.
Further, once you have used oil on any type of whetstone, water may no longer be used as a lubricant, as it will be shed by the oil that has soaked into the stone and be ineffective in carrying away the swarf (metal abraded from the blade by sharpening).
Here is a FAQ on natural waterstones from an online retailer.