The chemical make up in tires can definitely leave staining in some surfaces like your wood floor which is likely covered in a clear urethane of some type. For a DIY fix think of this problem along the lines of a small scratch or stain in the paint/clearcoat of your car. You may want to try a MILD auto polish to try to gently polish out the stain.
Start with something very mild - DO NOT use use something abrasive like comet cleaner or magic eraser, especially if your floor has a shiny/clear finish. Those things will abrade the shine right out of the clear coat and leave you with a dull spot.
Start with a test spot in a corner or out of the way place of the floor and see how the polish you use will work with the floor's finish. Apply a quarter sized drop of polish on a microfiber cloth and rub/polish the spot. Rub on the spot for 1-2 minutes and check it frequently to see how the finish is holding up. You don't want to rub away the floor's shine causing the finish to look dull or different. If its rubbing off the shine, stop- and don't worry, because this should be a test spot in an out of the way place.
If it looks ok, move to the spot and give it a go. Apply a quarter sized drop of polish on a microfiber cloth and rub/polish the spot.
When I say automotive polish I mean something with a mild polishing agent to be used on clearcoat auto paint. Maybe you already have something like this on hand. These are some inexpensive items widely available as an example:
