What is causing my knees to radiate heat?

Inflammation: Warm joints are typically caused by inflammation, a healing response mediated by your body's cells. An injured area attracts healing white blood cells, macrophages, etc to start the repair process. Classically the process includes redness, swelling, tenderness, heat, and pain as the blood supply increases as a response to bring those cells after injury, gout, infection, etc.

Many possibilities: An increase in the temperature of the knee can be due to many things such as gout, infection, inflammation of the bursa, or due to tendon, cartilage, or bone abnormalities. Have it evaluated by a professional and get the appropriate treatment.