To what degree do airline ops specs preclude hand flying?
There are no Ops Specs that preclude hand flying, however company policy and operational considerations do in specific situations:


Level flight within RVSM airspace 
Cat III approaches (and I believe Cat II approaches) 
High workload, critical phases of flight (company specific policies here) 


The FAA even came out with a SAFO not too long ago recommending that pilots do more hand flying during periods of low workload (and during training) with the goal of maintaining hand flying proficiency.