I will file this post away as one more reason why HDMI is as often more of a headache than it is worth. It sounds like a plasma issue to me. The plasma isn't looking for HDMI signal constantly, but only looks for it once.
Not sure of a fix unless there is a new HDMI board that can go into the plasma. I would check with the manufacturer to see if they are aware of this as an issue.
It may be worth it to pick up an HDMI DVD player just to confirm that this issue is consistent among products and isn't isolated to your Dish box.