So the IP Office screams every time a call comes in with no caller ID received. Avaya finally decided to put in a workaround, and it's super quick and easy to implement!
The workaround is only available in 9.1 (and up, I suppose).
Go to the NoUser user and click on the Source Numbers tab.
Add the source number SUPPRESS_ALARM=1
Merge your changes and that's it! The NoCallerID Notification will be suppressed for System Monitor, System Status, Sys Log, SNMP Traps, and e-mail notifications.