This section of the novel comes in Chapter Eleven, as Jonas completes his first day of his assignment. Crucial to understanding why the old man looks said is what has just come before. Note what Jonas says about his experience of having sunburn and feeling its pain:
"It hurt a lot," Jonas said, "but I'm glad you gave it to me. It was interesting. And now I understand better, what it meant, that there would be pain."
In response to this, the old man doesn't say anything. Let us remember what happened to Rosemary, the Receiver of Memories before Jonas, who couldn't take the pain and suffering of memories much, much worse than sunburn, and volunteered to kill herself. Presumably we can infer that the old man is looking "drained, and a little sad," precisely because he knows what Jonas doesn't: the pain from memories such as war and dying are infinitely worse than a bit of sunburn, and Jonas has to face all of these memories, too.
The old man, who tells Jonah that his name is now the Giver, feels sad because in addition to imparting fun memories to Jonah such as what it feels like to ride a sled down a snowy slope, the Giver must also impart painful memories to Jonah. On the first day of training, the Giver has Jonah receive a mildly painful transmission, which is the experience of a sunburn. Jonah doesn't even realize what it's called, but he knows that it's painful. However, "the pain kept him from grasping it." Jonah has never experienced any pain before, even something as innocuous as a sunburn.
The Giver is sad and weary because he has to bear the painful memories for the entire community. He also fears having Jonah receive memories that are extremely painful that Jonah has never experienced before. The last Receiver of Memories could not handle the pain of her position, and the memories were released into the world. The elders want Jonah to absorb all the memories so that only he has the pain. The Giver is sad because he knows that Jonah's training will be long and painful.