I have no experience with Fanuc robots but a quick google search shows they are of a 5 or 6 axis style robot.
If you can touch the robot when there is power applied and there is a chance the robot can execute its program then this is a very poor design of safety robot cell and needs immediate rectification.
The 6 axis robots I have worked with (Kuka, Epson, Staubli) all needed to be moved by teach pendant/HMI after a crash. You should not over-ride the brakes as stated above.
With the exception of maintenance activities which of course should then be carried out by skilled, competent persons. The epson robot controller as an example has a D-Sub connector on the back, which you can inject 24V in to to release the brakes, I have had to use this when changing the servo on one axis for accessibility.
SCARA robots on the other hand, these can be pushed whilst the cell is in a safe state (i.e. a Castell or similar key is removed before entering the cell and stays on the person within the cell). The Staubli/Bosch-Rexroth SCARA's come included with a button on top to release the 4th (Z) axis, this can be used both to clear after a crash and for teaching (move by hand and then enter "here" on the pendant).
TLDR: No.