CSULB recycling doesn't take rubber products. I cut the metal valves out of inner tubes and recycle them at CSULB by depositing them in the scrap metal bin.
My local recycling center also doesn't take rubber products - only the standard glass, plastic and metal (aluminum, copper) items.
Maybe your local car tire store can recycle the bicycle tires and tubes. My local car tire store used to take bicycle tires and inner tubes as a courtesy, but they don't do that anymore.
I checked the Los Angeles County Department of Public Works for the Household Hazardous Waste program. Rubber products (bicycle tires and tubes) are not collected.
So, if a local car tire store can't (or won't) recycle bicycle tires and tubes, you (we) unfortunately have to throw them away.