Question: A client of mine has 2 PLC processors and wants the newer PLC to control a motor that the older one already controls. They are phasing out the older PLC but prefer to add logic to the newer PLC as time goes on (I know this sounds strange). My question is: Can I simply connect the output wire from the newer output signal onto the same terminal screw of the motor starter coil as the existing output signal wire from the older PLC? Both processors are Allen-Bradley ControlLogix5000 processors.

Answer: 
If the output cards are relay cards, and the common is linked between the two PLC's, then you should be OK.

I would recommend taking both outputs down to a diode terminal, just to be safe
No, to connect both to same motor coil contact is unsafe and would not be best practice. The correct way is to have both old and new ControlLogix 5000 processors on the same Ethernet network. Then on old controller make desired tag global consumable and on the new controller, add an 'OR" branch with old controller tag to rung of new controller rung outputting to motor coil contact.