The Giver does change through the novel. A lot of the changes are the result of greater characterization, as we learn more about him. Others are changes in his behavior and personality sparked by interaction with Jonas.
The Giver goes from being a shadowy unknown to a fully-developed character. For example, when we first see The Giver in the Ceremony of Twelve he is only described as the Receiver, an old man set apart from the others. We don’t know anything about him except that he is one of the elders. When Jonas is assigned the role as Receiver, this man becomes The Giver.
At first, The Giver seems harsh and distant. Jonas is wary of him. We learn that The Giver is the way he is because he is usually left alone, and not used to company. His position separates him from others, and while he is greatly respected he is also different so most avoid him.
Jonas breathes new life into The Giver. Although he is stern at times, he begins to open up to Jonas. They have many conversations, often about important and serious issues. He proves to have a gentle and even sensitive side, and we learn about his past. We learn that he once acted as Giver to his daughter Rosemary, and the experience caused her to commit suicide. This left The Giver wary and sad.
Through his relationship with Jonas, The Giver comes to realize he does not have to accept things as they are. He agrees to help Jonas with his plan to run away, and even collaborates with him in terms of providing him memories that will help him on his trip and helping him coordinate the best time to do it, as well as agreeing to help the community with the pain when the memories return to them. They both hope that this will bring change to the community, and put an end to atrocities like Release.
So you see, The Giver not only goes from being a flat character to a fully-developed one. His relationship with Jonas causes him to change.