To my knowledge there would be no problem with getting in the FIR (Far-Infrared-Sauna) sauna while fasting, it seems it would be a great time as you will be sure to have pleny of fluids and will not get dehydrated. :)
When sweating in a FIR (Far-Infrared-Sauna) sauna you sweat out 20% toxins and fat, 80% water (whereas a normal steam sauna is 98% water, 2% toxins and fat) and when you drink water you replace the fluid you have lost--but not the calories (fat) or toxins. Once your metabolism is revved you will sweat quite a few calories, how they calculate how many is a mystery to me.
A FIR (Far-Infrared-Sauna) saunas benefits are many...great for circulation, the heart, detoxing (mercury especially), CFS, Fibromalygia, just your whole body. I highly recommend them myself. :) Namaste, Caitlin Lorraine