Solution:
I was able to actually able to fix the problem! Turns out, when I was logged into WordPress, the browser was unable to serve one of the images associated with the WP admin bar, resulting in a 404 error. For whatever reason, that made the whole site break.
As long as the visitor is not logged in, all works fine (explains why I was getting the results above, as I was logged in when I viewed the site in Chrome, but was logged out when I did so in Firefox and IE).