What is the most important maintenance issue?

The single most important aspect of bagpipe maintenance is the issue to moisture control. An instrument that varies with respect to moisture can never be stable nor reliable. 

Water traps, desiccant systems and pipe bags must be chosen to be suitable for the climate in which you live and the proper considerations for maintaining an appropriate moisture level.

Moisture absorbed into a reed makes it both heavier and more flexible, so the native pitch of a wetter reed is different (generally lower) than that of a dryer one. If the pitch changes as you play, you'll have a problem.  If the level of moisture in the bag is too high, the reeds will become wet in the short term and will mildew in the long term.  (Mildew is evidenced by a black color on the reeds.) For this same reason, the pipe chanter should never be left inside the bag when not in use.  It should be kept outside the bag with a reed protector in place. I'd also recommend swabbing out the chanter stock after each day's playing to remove water or saliva that may be absorbed into the wood.  Swab out the drones and chanter stock on cold days.