I would think not. Most staining from Dyes is a result of the tubing being EVER SO SLIGHTLY porous, and the pigment molecules seep in. Pumps are made of plastics and metals and ceramics specifically designed NOT to be porous, so I don't think it'll be an issue.
However - certain types of stains (from Water Wetter for example) are made from a thin coating that is left on everything. this DOES affect your pump, as well as your waterblocks and evreything else.