What I think will happen in chapter 2 is that somebody will notice that Lina and Doon switched jobs. I think you may not be allowed to switch jobs. I also think that when the person finds out he will tell the mayor. The mayor might get so mad that he sends people to find Lina and Doon.
My other guess is that they switched their jobs. Afterward they become friends. When Lina is working as a messenger she might ring the doorbell to a house where nobody answers. She might go inside the house and look around. She might find the box with the map to escape the City of Ember inside it in a closet. She would probably show Doon the map. Doon would most likely want to tell everyone and try to convince them to try and get out of the City of Ember. He would try to get them to leave so they wouldn't have any more problems with the lights.
The mayor probably would not like Doon's idea because of his opinion of him from the assignment day. The mayor would try to convince them to stay in the city. If he couldn't convince the people to stay he would try to get the map from Lina and Doon so they couldn't escape. That is what I think is going to happen.