Jonas's future is left unresolved as he is left at the end of the book to look down upon the place he has searched for. The outcome is basically left up to the reader. Does he live? Does he die? The future is uncertain; however the literal answer to your question is no, Jonas does not die at the end of the book.
No one knows for sure if he had died or not. However, in the book, the author wrote that Jonas used the last of his strength to push the sled forward one more time. Therefore, it was not clear whether or not he died.
No, Jonas does not die in, "The Giver." He shows up at another book as a leader with Gabriel.
Ok, I want to make this short. Personaly I don't think so. People have their opinions. But there are two other books and Jonas is mentioned there. I have only read The Giver and it does not state if he died or not. Again it's opinion.