Garage door opener makes noise but doesn't move
It sounds to me like the motor is not starting. If you can manually lift the door, then I'd start by checking the motor capacitor.  If it's on the way out, that could explain the intermittent failure.  

You may notice physical damage to the capacitor (leaking, buldging, etc.). More likely you'll have to measure the capacitance, to make sure it's within a good operating range.

While the unit is open, inspect all the gears and sprockets.  Look for worn teeth, or gears that don't mesh properly. Also inspect the sprocket on top of the unit (you'll have to remove the guard), to make sure it's meshing with the lift chain/belt.