The problem is that a forge doesn't burn coal to heat steel. It burns coke. The coal that's burning is just to change it to coke and is around the outside of the fire.
I don't think you can coke tire rubber and burning it will be a nasty tarry mess. You would have to figure out how to separate the burning from the metal heating. Sulfur contamination is another big issue as sulfur degrades steel.
So is it possible? You could probably figure out a way to do so with a reverbatory furnace (look it up) and it would only be many more times as expensive and as difficult that using coal or charcoal.
You know it's possible to burn diamonds as forge fuel, just not "efficient".
BTW the recycled rubber is known as "crumb rubber" according to my father who was on the board for a tire recycling business.