After troubleshooting the issue and speaking with tech support, it turns out that the Terastation Live doesn't support UPS shutdown communication even though it has a serial port. What is the purpose of the serial port if that is the case? If it is not usable, I find its presence to be deceptive. I'm guessing that I'm not the only person to assume that the inclusion of the serial port implied that the device would work with my UPS. This is very frustrating.
Use the link under "Articles" on this page to learn how to "Activate UPS Support on TS Live". You will need to install ACP_Commander first (instructions also on this link) to get Telnet access to the TS Live.
You will find other useful TS Hacks on this page as well.
Good luck.
Unfortunately, modifying the device's operating system in any way can void your warranty. The warranty states:
"If the Buffalo product malfunctions during the warranty period, Buffalo Technology will, replace the unit, provided the unit has not been subjected to...non-Buffalo Technology authorized alteration, modifications or repair."
The full text of Buffalo's warranties is located here:
http://www.buffalotech.com/files/Storage-Multimedia%20Warranty%20Statement.pdf
UPS support is not included in the Terastation Live, but I will forward up a feature request.