Core Issue
In an attempt to build a replacement Cisco CallManager in preparation for the upgrade to Cisco CallManager version 3.3, another Cisco CallManager 3.1 has been built and a backup from the production server has been restored. However, test IP phones on this particular server do not work, and immediately go to fast-busy when going to off-hook.
Resolution
When simulating a lab environment, it is best to use the same versions of software that the production environment has. In this particular instance, the production server is running Cisco CallManager 3.1(2c), but the lab server is running Cisco CallManager 3.1(4b).
The restore of the 2c database is not going to work on the 4b, and if it did, unforeseen problems are likely to occur.
Downgrade the lab server to the same Cisco CallManager version as the production server (the correct way to test production servers). In this particular scenario, the downgrade to Cisco CallManager 3.1(2c) resolved all of the issues.