Technically such a thing can be done (would've happened if the TiVo suit turned bac), but it is not a practice or policy of which I am aware at Dish.
I am almost certain in your case the answer is NO, as if it is a leased box, then Dish would want that DVR back for those who want a DVR (they are more expensive to manufacture) and furnish you with a non-DVR box. If your DVR is owned, then I am equally certain the answer would be NO, and they would advise you to buy a non-DVR box. You own the DVR; it's your problem, not theirs (that is how they would see it).
It sounds like what you need to do is bump your DVR recordings to VCR tapes or DVD's to watch, then get a non-DVR box.