What is the material on top of the chips in this logic board, and should I leave them?
They're to protect the IC package against mechanical impact. Total height of such an IC may be less than 1 mm, so you can imagine the top of the package is only a couple of tenths of a mm thick. A sharp object will go right through it and destroy the die.  

Looks overkill to me. The photo shows bubble foil too, and there's usually a few cm between board and the carton box. If during handling a sharp object can penetrate 3 cm deep into the carton, and still have enough energy to break the IC's package your handling process really needs a critical review. 

Also the rest of the board isn't protected. That same sharp object may scratch the PCB, cutting traces, or chip off a small SMT capacitor.

You can throw them away.