Yes you need to install new saprouter in new system.Usually it be one sap router.
you don't have to install saprouter in each system.
When you install new systems..i am hoping you will keep old internal IP addresses for the systems. If you are changing the Internal IP address of systems too then you have to change the saprouttab entries too
SAPROUTTAB will be replaced in only saprouter which is only one.
Hardware key comes into picture when you migrate your systems to new hardware. and it does not impact saprouter.
Regards
Anand