I guess it might be possible but both materials, the tires and the covers, are made from rather inert materials. With the tire you're looking at probably synthetic butyl rubber and coal black as the main ingredients and the cover would be some type of spun plastic, a long chain polymer like polyethylene or nylon. There is just not much chemical affinity between the two to have a chemical reaction. If anything, I would assume the coal oil in the tire could cause the cover to change.
The cover discoloration may be from the elongation or compression of the cover material, causing it to hold dust. Could also be a solution was applied at the tire dealer. I know most use a soap solution to mount tires. It may have tire dressing on it as well.