Fatigue is an effect that occurs when a member has stayed in the fallout shelter for more than (around) 20 days which cannot go away. It does not influence the gameplay too much and does not mean that the player shouldn't send them to scavenge. It is a normal effect to have. Mutant Mary Jane is immune to this effect.
The character that becomes fatigued starts to look slightly tired and dirty. It does not interfere with the game too much. Timmy will have slightly blood shot eyes and a cut on his face, Ted will look dirty and grumpy when his beard grows to its maximum length, he will also have quite ripped clothes, Dolores will look dirty and tired, and Mary Jane will look dirty and slightly tired. Dolores looks similar to when she is tired. All the fatigue effect does is it just decreases the amount of objects that a member can obtain when sent out to the surface to scavenge.