The best way to remove bike tire marks from carpet is by using a solvent such as Simple Green Carpet Cleaner, or mineral spirits. Always test a cleaning solvent in a small hidden area to make sure it does not damage your carpet.
Apply solvent to a soft bristle brush, and gently work it into the tire marks without scrubbing. You may need to allow the cleaner 10 to 15 minutes to loosen rubber particles from the carpet. Blot at the marks with a clean damp towel, followed by a clean dry towel.
Once the marks are removed, you can clean the solvent off of your carpet by using some weight shaving cream. Follow this up by sponging cold water on to the area, and blotting up all excess moisture with clean dry towels.