Why are the leading edges of the 787's horizontal and vertical stabilizers not painted?
Aircraft stabilizers (both vertical and horizontal) usually have metallic leading edges without paint or anti-erosion paint system. This can be seen in the following figure for Airbus A320 family.



Image from Airbus Structure Training Manual

For aircraft with composite stabilizers, leading edge erosion is arrested by bonding metal to the leading edge, or making the leading edge using metal, as in 787:


  The .. leading edge assembly, ..., with an aluminum edge and a fiberglass and paper-honeycomb skin.


In such cases, (as @Todd points out), the leading edge is usually left unpainted in order to prevent paint erosion. This method is used in the leading edge of the main rotor blades of a number of helicopters. In such cases, either the bare metal is left as is or a protective film is used to combat erosion.