You may have a problem that is existent in a small percentage. That doesn't make you are anyone else a liar. What we do on this forum is try to confirm problems as reported by ONE person and see how many on this forum have the same. I can tell you that for the TWO MT2's I have, and use often, I have had NONE of those problems, and I even checked for excessive heat after your original post, leaving it tuned to an OTA, and I really couldn't tell the difference. I think it may have gotten hotter, but, if so, not alarmingly so. So, the problem has not occured for me in the approx. 2 years I have had the MT2's and used them often for OTA. Several others have posted the same "no problem here." What does that mean? That you and at least one or two others do experience this problem means it is occurring in very few circumstances, which means it is very difficult to figure out the problem. All one can do in your case is try all the different scenarios to get the OTA tuner to function reliably, and that takes time and PATIENCE. You may have to swap the box and/or the tuner and who knows what else. But I certainly wish you luck and hope DISH can accommodate you in RMA's or technical visits.
TiVo also uses the very same MT2 for its older Series 3, HD, and HDXL and for its newest model, Premier and Premier XL, and NO problems of MT2 overheating or seeming excessively hot reported on the TiVo forums. It could be how Dish has things laid out as opposed to TiVo. Keeping an open mind, but if others are not having a problem accessing OTA or recording OTA, then it isn't a problem.
On the other hand, I can report LOTS of tuning and signal strength problems due to our STUPID ATSC, as opposed to DVB-T. UHHHH! I'm dealing with another ATSC PITA right now. It drives ME NUTS.
Please let us know when you have found the solution so that others don't have to go the long road you are blazing for the rest of us.