Making a tired face is a facial expression where the person just looks really tired, a little sad even. Imagine someone, who is working very hard on his job everyday and gives his best at all times. When he comes home, he's tired and thats the way he looks: he's making a tired face. However, in this context, it sounds to me as if the man is acting as if he was. "making a tired face" is usually an expression people make, when they worked hard for something. "Just having to make a tired face", means trying to seem like someone who did.