There are three main reasons why the Giver does not go with Jonas.
The Giver realizes that the community needs to change and that Jonas leaving, and him staying to help, is the way to do it. If he did not stay, the community would be too overwhelmed by the memories to make changes. They would not have the wisdom to make intelligent choices based on those memories, and that is why they need the Giver.
The Giver has a sense of duty to the community and he wants to stay and help them deal with the memories they are about to receive. He is the best equipped to do this, as he will be the only one in the community who has ever experienced the type of pain they are about to go through. He knows that if he were to leave with Jonas, he would be completely abandoning the members of the community in their greatest time of need.
The Giver is too old to go on the trip with Jonas. He is very old and very weak, as we have seen in the memory sharing sessions. He would not be able to travel "elsewhere," and he really doesn't want to. At this point in his life, he is looking forward to death. His life has been hard and, as he says, he wants to be with his daughter (she died and he is looking forward to re-uniting with her).