Move disk to new hardware?
Checking the specs of the the MacPro1,1 you should be able to upgrade to 10.7.5 if you upgraded it to at least 2 GB of RAM and you don't need Rosetta (for PPC-apps).

Since the preinstalled system on the MacPro 3,1 was 10.5.1 and both are equipped with a SATA (3 Gb/s) internal HD-interface, there shouldn't be any problem to move the "old" disks.

Your disks won't be updated with anything, except you initiate it yourself.