Tires are made of 3 materials, the rubber, cloth plies of various materials such as nylon or polyester, and beads, typically made of steel or kevlar. It's very difficult and expensive to recover these materials for recycling. Even auto tires which have more to be salvaged, are often not recycled for raw material, but are simply repurposed for all sorts of odd jobs.

Unfortunately, bike tires don't have enough material to warrant the cost of separation, and there's not any decent repurpose applications so bike tires end up in landfills, regardless of any well intentioned detours past the recycling center.