Do you have Gout? (long shot)
Do you eat fruit or fructose drinks when riding? They can exacerbate gout symptoms.
Does the burning feel like a blow torch? That is what gout inflicted my left big toes felt like starting Transam. An electrical japping burn would be more nerve pain but I don't ever remember reading of someone with nerve pain showing in the knee. Hands, elbow, shoulder, neck? yes.