How do you remove the yellow on your tile caused by the bath mat?

Unfortunately, it's likely you can't. The rubber backing on the rugs is the culprit. When the rubber backing on the throw rug gets warm, the resulting chemical reaction can leave an unsightly stain that will be clearly visible on the vinyl flooring. Throw rugs located in direct sunlight are especially problematic. Although the stain is most often yellow, the color of the stain will vary. Stains caused by throw rugs are sometimes impossible to remove, but a simple fix may resolve the problem. Also, other things can react and cause stains. Be careful when placing items like flowerpots, rubber wheels, anything made of plastic, etc., on a vinyl floor.

You can try a baking soda paste. Remove the throw rug from the stained area. Sweep or vacuum any loose dirt.

Make a paste of warm water and baking soda. Place baking soda in a small bowl, then add enough warm water to make a paste with the approximate consistency of toothpaste. The amount of baking soda needed will depend on the size and severity of the stain.Wipe the baking soda paste over the stained area. Allow the baking soda paste to remain on the stain for several hours, or overnight.Scrub the stain, using a washcloth or a plastic scrubber and warm water. Rinse the baking soda from the stain. Repeat, if necessary.

Make sure you have cleaned all foam backing off the tiles with GooGone. Then clean the tiles with bleach and water. If the yellowing stain is still there, pour pure bleach over the affected tiles and let it sit at least three hours. Then sponge or mop up the bleach and rinse the floor. (I use a damp mop and mop the whole bathroom floor, then wash out the mop in the tub, and use the damp mop to rinse the floor.) You could also try baking soda and peroxide. The stain should at least be better. And buy a different kind of rug in the future. ☺️

I would scrub it again with the Goo Gone. Good chance there's residue, still. I hate working with bleach, so I would then use a paste of baking soda and peroxide. Good luck!

Try Mr Cleans Magic Eraser. It is good at getting most stains out for me

I can't use any rug with the rubber backing as my cats think it is a sand box. It has taught me that the backing is not very healthy either.