Restoring Backups on a Cold Spare

Will restoring a backup on my cold spare change the system IP under network settings to the IP of the system off of which the backup came (my live system)?

No.  Network settings are not affected by restoring backups.
When restoring a backup on a system, always review the settings in IP Configuration to ensure they are accurate. This includes values such as Hostname, which are neither recorded during the initial Networking Setup in the Console nor saved in the backup file.