My humble fix for a sweaty toilet
We used to have an on demand water heater located in the same area as our toilet. It had a massive 5 inch vent with a standing pilot light. You could sit on the toilet and use the heater as a sonic periscope, listening to all the sounds in the neighborhood. We finally replaced that beast, as it was finicky, made it difficult (if not impossible) to get just a small stream of hot water, and we finally had an area to place a full size electric water heater.
One of the downsides to this, we discovered, was that there was no longer a constant flow of air flowing past the toilet tank. This meant that the cold water coming into the toilet condensed on the outside of the porcelain tank, causing large droplets of water to fall all around the toilet, creating issues with the wood floor of the bathroom (long story). So I decided to fix this problem.
My solution was to line the inside of the toilet tank with half inch thick neoprene foam, held in place with 3M VHB adhesive strips. I covered the bottom of the tank, cutting around all the vital bits, and lined the wall, going up a half inch above the water level. Problem solved.