I think they will die, but nothing devastating or tragic. I think Amy and Rory will die of old age when the Doctor comes back for them, whenever that may be during the seventh series. As many will know that they went with the Peter Pan themes and connections for the Eleventh era between the Doctor and Amy (him being the Peter Pan to her Wendy), and we all know how the story ended for Peter and Wendy. There's a pattern with the Doctor and Amy as well, how he keeps losing track of time streams and missing days, months, years to coming back for his Amelia Pond. Even in the recent Christmas special two years had passed since Amy and Rory learned the truth about the Doctor not really being dead, and while he was intentionally staying in the shadows he could have visited them. He even asked how long it had been since he had left them. Of course we all know that despite the gaps he always comes back, but what if the next time he's too late?
Either he comes back to find them of old age, on their deathbeds or having already died and left the Doctor a note, knowing that he would come back looking for them as he always does.
This is what I interpret how depressing their departure will be. Because the Ponds have always had the Doctor in their lives, ever since they were children, even without realizing they were connected to the Doctor and the future adventures they would have together (i.e. Mels) I know that Karen Gillan has mentioned that she would prefer Amy to die, and immediately we think of the worst. But it doesn't have to be something horrible and tragic, still depressing but more bittersweet because we all knew this day would come sooner or later, especially the Doctor. Which out of all the companions he's cared about, I wholeheartedly believe the Ponds are the first he truly considers his family, so this will no doubt be very emotional for him, particularly since the Christmas special has him being all humany-wumany with "happy crying" by being invited into their home for Christmas dinner, even after two years of not being there, he is still welcome by them, and always will, and that? That to him is his home, these people are his family, his home, those he loves and cares about and the feeling is mutual and their departure, however it happens, will definitely crush him, even though he probably knew it would happen one day anyway, them being human and him being Timelord and things being all wibbly wobbly in the time streaming sense.
So, yeah. This is how I envision their end to be. And honestly, I can't think of anything more fitting for their story to end than in this manner. It stays true to their characters, their journey with the Doctor, and the Peter Pan-esque theme they went for in telling this story from the beginning. It would also be just the Moffat thing to do to tell such a heartbreaking end, but in an uplifting way.
Overall, I think the Christmas special really got to me because we know what is coming, and seeing such a heartwarming episode brings the heartache of what will come in the next series (along with the foreshadowing, "What's the point of them being happy now if they're going to be sad later? The answer is, of course. Because they're going to be sad later") It has just made me very hyper-aware that the next series will be the end of the Doctor and the Ponds and while I know this must happen, it doesn't make it any less painful. :(
STILL, HUGS! HAPPY CRYING! FAMILY! THE GLORIOUS PONDS AND THEIR RAGGEDY DOCTOR!