If you are moving the entire network contents, router, SB , etc. You shouldn't have to reinstall the SB. Your router will have a new network IP, but I don't think that would affect your SB operation. The Finder ID will not change. Slingbox should be able to find your ProHD once your router is back on the Internet.
If you change or use a different router, I would run the SB installation program again. There are port forwarding rules that need to be setup within the router for the SB device.