I haven't had much luck in finding anyone who recycles old bicycle tires. There are some companies that accept them for recycling, but are probably not convenient for most people to use. Simply dumping them in a landfill partially negates the otherwise environmentally-friendly nature of bike riding.
I contacted TerraCycle on Twitter and asked about bicycle tire recycling and was told that they had tried partnering with bicycle companies on this issue, but had not much luck.
However, I did find numerous listing for "bicycle tire belt" on a marketplace website, called Etsy, that features handmade items of all kinds for sale.
Here is a picture of a finished belt from one store, so you can see what they look like
But, if you'd like to take a crack at it yourself, there are online tutorials for making belts from old bike tires. You can find those tutorials here and a better article is here.
I think that old mountain bike tires would probably make better belt material than road tires, due to their thickness and durability. Plus, they look way cooler.
I have some old tires that I've been hanging onto in the hopes of finding a recycler close-by, but I've had no luck in that regard. I may try my hand at making a belt or two. If you really wanted to find a way to offset the environmental impact of discarding your tires in the landfill, you now have a choice between making them into something useful or you could buy a belt from one of the Etsy shops I've listed above.
If offsets can work for greenhouse gas emissions, I think they can work for bicycle tires, too. I was never very good at shop class in school and I don't have the necessary tools to cut and shape the belts properly. Personally, I have no idea how many cyclists and other people would either buy or make these things. But, it could fill a niche for people who want to be "trendy". After all, just ten years ago, who would have thought that fixed gear bicycles would become as popular as they are now?