I found some other forum posts about this behavior and the root causes seemed to be 1. lack of permissions and 2. inability to load JS from the mt-static directory.
I verified with Firebug that all of the JS files loaded from mt-static correctly and then reset my and my other author's permissions. This got everything working.
I then logged in to answer a question for my other author about editing templates and got the grayed out/unable to type problem again. As soon as this happened to me it ALSO happened to him.
Is there a known issue where two people are at the Edit Template screen at the same time? I logged out and had him logout/login but that did not fix the problem. He also tried doing a full site rebuild but that did not fix the problem either.
Tested in Firefox 2/IE 6.