You didn't mention if the burning sensation is in the front or back of your knees. Behind your knees: Could be a popliteal cyst, however they are not common on both sides at the same time. It could be sciatica, although the same thing applies here. I would guess something to do with your lowest back vertebrae. Same with the front of your knees. Lumbar or sciatic disc pain can radiate around to the front of the leg, and can happen to involve both legs at once depending on how much of the disc is involved. I'd see your Dr.

Best wishes,
Ladywitch