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). 

Why will The Giver not go with Jonas?

After the Giver shows Jonas the video footage of his father lethally injecting an infant during a routine release, Jonas is horrified and appalled with his community. Jonas refuses to go home and ends up spending the night in the Annex with the Giver. That night, the Giver and Jonas come up with a plan that will alter the community's future for the better. Their plan requires Jonas to fake his accidental death while he secretly escapes the community and travels to Elsewhere hidden in a transport vehicle. In chapter 20, Jonas begs the Giver to leave the community with him and accompany him to Elsewhere. The Giver responds by reflecting on how the community reacted to the loose memories after Rosemary's death. The Giver tells Jonas that he must stay behind to give the citizens wisdom and help them cope with the difficult memories after Jonas leaves. The Giver fears that if he leaves with Jonas, the citizens will panic and chaos will ensue, which is why he decides to stay back.

Why will The Giver not go with Jonas?

When Jonas decides to leave, he pleads for The Giver to come with him.  The Giver responds that he is old and not strong enough for the journey.  He also says that the community will need him.

After Jonas leaves, they know he will face harsh conditions.  Long hours of running, harsh weather and lack of food would be very hard on The Giver.  He would likely slow them down.  Gabe is enough of a burden, since he too needs to eat and be kept warm and cannot fend for himself.  The Giver has also given Jonas many of his memories, possibly leaving him not enough to sustain himself and them.

The main reason The Giver cannot leave is that when Jonas leaves, all of the memories he has will be released back to the community. Chaos will ensue unless someone is there to help them.  With both Jonas and The Giver gone, all of the memories of pain and suffering would go back to the people.  They would be confused and not know what was happening to them, and without The Giver there would be no one to help them through it.