Chemical Reaction
In most cases, when the floor under a rug turns yellow it is due to a chemical reaction. Rubber-backed rugs are to blame. A chemical used in the rubber to keep it soft and flexible reacts with vinyl flooring. Eventually the chemical transfers to the floor underneath, permanently discoloring it. Unfortunately, once the reaction takes place there is no way to remove the yellow stain. Options are limited to replacing the flooring or keeping it covered with a rug.
Aging
When a rug has been in place for a long time it is common for the flooring under the rug to be a different color than the area exposed to sunlight and traffic. Many times the spot under the rug will appear lighter or cleaner than the surrounding floor, but in some cases it will appear to have yellowed. This can be from aging or from fading of the rest of the floor. In the case of linoleum it is likely from failure of the components to naturally react with sunlight. Removing the rug and leaving the area exposed to light and traffic will usually solve the problem, but it may take time for the entire floor to appear the same color.
Prevention
As the saying goes, an ounce of prevention is worth a pound of cure. It's much easier to prevent a floor from turning yellow than to try and eliminate the stain after it appears. Avoid using rugs, mats or non-slip materials that have rubber or PVC backing that comes in direct contact with the floor. Rugs that are made only of fabric are ideal. To prevent slipping, look for fabric or fiber pads that are designed to hold rugs in place. Also, move rugs to different locations on a regular basis to prevent discoloration due to aging.
Considerations
Rugs are not the only items that can cause yellow marks or spots on flooring. Any item with a rubber base that comes in contact with the floor could cause yellowing. Common household items that may be a problem include stepping stools with rubber feet, rubber wheels on carts or rolling shelves and toys with rubber parts or wheels, like a child's tricycle, that are allowed to sit in one place on the floor for any length of time.