At the end of Harry Potter and the Goblet of Fire we see for the first time an actual Dementor's Kiss given to Barty Crouch Jr.
We know by fact that a Dementor's Kiss doesn't actually kill the person to whom it is given; it just takes away his soul, leaving him soulless but still alive.
What was the fate of Barty Crouch Jr. after he was given the Dementor's Kiss?
The last time Barty Crouch Jr. is mentioned is in Goblet of Fire, where his soul is sucked out by Ministry Dementors, as mentioned in the question. To the best of my knowledge, his fate has never been mentioned again in any official, canonical, or licensed Harry Potter work. He merits a passing line in Order of the Phoenix (mentioning past events only), and that is about all.
It seems most likely, though, that his body would have been taken to St. Mungo's Hospital and kept on whatever the wizard equivalent of life support is. He would not have had anyone to say otherwise, regardless of his preference. Of course, it is possible that standard policy in the wizarding world is not to keep victims of the Kiss on life support, particularly if they are convicted mass murderers, so even this is uncertain.
Ultimately, we simply do not know. In the end, though, whether Crouch Jr. is technically alive or not, he is effectively dead for all plot purposes.
I don't think he could have started a new life somehow because once you get kissed by a Dementor you cannot live like a functioning, ration person.
"You can exist without your soul, you know, as long as your brain and heart are still working. But you'll have no sense of self anymore, no memory, no...anything. There's no chance at all of recovery. You'll just — exist. As an empty shell. And your soul is gone forever... lost." —Remus Lupin (Prisoner of Azkaban)
But according to Harrypotter.wiki (http://harrypotter.wikia.com/wiki/Dementor%27s_Kiss) its not been made clear as to what happens to the soul once its sucked by Dementors.
I have wondered about this myself. It seems a person is able to live without a soul, but it's left to debate how. In Professor Snape's classroom, when he is the DADA teacher, there is a picture of a victim of the Dementor's Kiss, showing the person just sitting, hollow. It seems there may be a similarity between imprisonment in Azkaban and suffering the Kiss, although, upon release, the prisoners seem to come back to themselves..whereas there is no returning the soul back once the Demetors have taken it...leading to further questions..where does the kissed soul go..and a connection to Horcruxes, but a maimed and split soul by dark magic can be made whole again by remorse, but the pain can kill you, however yoir soul is once again whole and able to 'move on'...very interesting subject..possibly, Rowling left this open on purpose :)
