Are pilots required to obtain ATC clearance or permission to fly helicopter routes and transitions?
According to the FAA's JO7210.3AA Facility Operation and Administration (chapter 11, section 5):
Pilot adherence to charted helicopter routes and the recommended altitudes or flight ceilings associated with them will normally be voluntary. However, controllers may assign charted routes and altitudes and expect or request pilot compliance with them, provided such procedures are called for in specific FAA−operator Letters of Agreement, or are necessitated by traffic density and/or safety considerations; controllers also may restrict operations within designated operating zones when requested by local law enforcement officials and the restriction would not adversely affect other aircraft operations.
In other words, it's mostly a voluntary thing but pilots should still be aware of them and prepared to follow them if instructed.
And you have to consider the airspace as well: large parts of some routes are in uncontrolled class G airspace, e.g. the PALISADES coastal route, west of Santa Monica. ATC doesn't control traffic in class G, so asking for permission wouldn't make sense.