What did Jem learn from his encounter with Mrs. Dubose following her death?
This is a good question. But we need to be nuanced here. What Atticus intended to teach Jem and what Jem learned might be different.
Here is the context. Atticus had Jem go to Mrs. Dubose's house to read to her, as a punishment for destroying her flowers. So, Jem, as a dutiful son, went to Mrs. Dubose's house to read to her on a regular basis.
After Mrs. Dubose died, Atticus pulled Jem over and told him why he wanted him to read to Mrs. Dubose. He was trying to teach Jem a valuable lesson about courage. So, Atticus explained to Jem that Mrs. Dubose tried to overcome her morphine addiction before she died. She knew it would be hard, but she did it anyway. For this reason, in Atticus's estimation, she was the bravest person he ever knew. Therefore, he wanted Jem to see what bravery really meant.
Here are Atticus's words:
I wanted you to see something about her—I wanted you to see what real courage is, instead of getting the idea that courage is a man with a gun in his hand. It’s when you know you’re licked before you begin but you begin anyway and you see it through no matter what. You rarely win, but sometimes you do. Mrs. Dubose won, all ninety-eight pounds of her. According to her views, she died beholden to nothing and nobody. She was the bravest person I ever knew."
Jem picked up the candy box and threw it in the fire. He picked up the camellia, and when I went off to bed I saw him fingering the wide petals. Atticus was reading the paper.