The MG34 fires from an "open Bolt" The trigger would need to be pulled for the weapon to fire.
The MG34 is selective fire. The trigger has a toggle action to it. The upper half of the trigger is for semi-automatic fire, and the lower half is for full automatic fire.
It still might not fire as a '308 round is 51 mm in length and the 7.92 round is 58mm in length. If the 308 is pushed all the way into the chamber, I seriously doubt that the firing pin would be able to reach the primer!!