It is not revealed
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.