Though they provide warmth and comfort in baths and kitchens, rubber-backed area rugs can also leave permanent stains behind on vinyl and linoleum floors. Here's why: antioxidants, added to the rubber during the manufacturing process to prevent the rubber from drying out, react chemically with the sheet flooring. This chemical reaction results in a permanent color change. So, unfortunately, what you are seeing is not a stain that can be removed, but rather a physical change in the color of the flooring that cannot be reversed.
Vinyl and linoleum flooring manufacturers often warn of this condition in their usage guides. For example, the following paragraph appears on the website for Armstrong Floors:
"Place a walk-off mat at outside entrances to reduce the amount of dirt brought into your home. We do not recommend the use of rubber- or latex-backed mats because the chemical (antioxidant) used to keep the backing from becoming brittle can permanently stain your floor. We suggest a non-staining vinyl-backed mat or a woven rug that is colorfast. Most of these products are identified "colorfast" by the manufacturer. All Armstrong floor care products have been specifically developed to care for Armstrong floors. You may purchase Armstrong floor care products at your local flooring retailer."
So, after vinyl and linoleum floor stains have set in, your only options are to completely replace the flooring or just buy a bigger (and non-staining) area rug!