A 15 year old boy's total bilirubin is 0.8 and direct bilirubin 0.4. The total bilirubin is in range but direct bilirubin is slightly out of our lab range [0-0.3]. What might be the possible causes?

Meaningless. Difference. No clinical correlation.
High direct. If the direct bilirubin is high, i'd doubt gilbert's & my first concern is cholestasis from some medication you are taking. In fact, some of the macrolide antibiotics have been known to do this. It's not dangerous right now but you'd do best to get with your physician & see if dcing your rx settles this down & if so what other antibiotic might we use instead. Good pick-up.
Computers problem. This a problem I have whenever i get a result that is sent from a computer that has limits built in the old saying garbage in garbage out applies here the computer can not give shades of gray (no pun) and say this within normal bounds or possible lab error do not worry and follow the dermatologist's plan.
Gilbert's? Could be nothing or can be a congenial problem called gilbert's ( pronounced jil-bher's) . It can also be caused by gallstone disease. Follow up with your doctor.