The handlebars does not need any special maintenance. Periodically (every 500 km of road per 200 km MTB, however, before each race) should be monitored for tightening the screws, of the stem and both shifters.
Creases in carbon fiber you should carefully inspect the closing point of shifters and stem, subject to cracking. Same attention in the integrated handlebars. Then eventually replace the handlebar tape and knobs, in this case, each behaves as it sees fit.
A buyers' guide from Professionals Cyclists
Both the Pro mechanics and the same Pro (road and MTB), when they can freely choose, always choose the handlebars triple butted aluminum, the reason is very simple. A triple butted aluminum handlebars weighs just over a carbon fiber, on the other hand is very reliable, good resistance to impact and does not suffer when you shake the strong shifters and stem. Find just the comfort that you are competing with a good pair of knobs (MTB) and often a tape or gel (road). Everything else is superfluous ... ..