Removing Water Stains from Barn Board Paneling

The mineral oil has coated and seeped into the wood and the oil itself stains and it's not drying because oil is a lubricant. You could try sanding the wood to possibly even out the coloring and if you can even it out enough with the sanding to please you I suggest using a wood protectant like is used on wood decks in the future to keep the water marks from reoccurring. My personal opinion is that the beauty of barn board is in it's natural weather aging and, believe it or not, there are people out there who pay oodles of money for weather aged barn boards.