If you (or your teenager) have left black peel-out tire marks in the driveway, you know that time and rainwater alone aren't going to get rid of them. Tire marks are generally laid with friction and heat, which means the rubber has embedded itself into the surface of the concrete or asphalt and won't come up easily.
Try increasingly aggressive approaches for cleaning it. When you do, consider getting the surface sealed or re-sealed, as that will make it easier to clean next time.
Soap and Water
Sometimes the simple solution is the best one. Use a household cleaner mixed with hot water and applied with a thick, heavy bristle brush.
Let the suds sit for a little while to soften up the material. Scrub with the bristle-brush, add some more of the cleaner-and-water solution and scrub some more.
Even if this doesn't get the marks off all the way down to the surface, it will likely get some of the top layer of it off, making the rest of the removal easier.
Trisodium phosphate (TSP) is a powered chemical that, when mixed with warm water, is one of the most effective cleansers available. It's hard on skin, so make sure to wear thick rubber gloves.
Scrub and wet the stain alternately as with cleanser, working off as much of it as you can. It's especially important that the TSP be rinsed very well off the surface at the end and into the storm drain to prevent animals or children from coming upon it.
The last resort is to rent a pressure washer. This is a compressor attached to a water tank that shoots the water out at extremely high velocities.
Used in conjunction with a good degreaser, it will get the tire stain out; the problem is, it may also take the finish off your concrete, or pull chunks out of your asphalt, so you need to find the setting that's powerful enough without being too powerful. When you rent it, talk to the home improvement store about what you're using it for and what the recommended pressure and nozzle angle for your model should be.
(Rent one that can go to 3,000 pounds per square inch, to make sure you have enough power, but it's unlikely you'll use it at full strength).