No, they don't. They just run hot. There are a ton of little ASICs behind the flat plane running the various antenna bits required for the beam forming (directing the beam without physically moving the dish, once it finds a spot it mostly stays put), and they all put out some good heat. Plus, the high frequency RF involved isn't the most efficient in terms of antennas.
Yes, they're warm. No, it's not a dedicated heater mode.