Craft hangs when updating entry
It turns out that I was running a dev release of PHP 5.6 from a PPA and not the official Ubuntu release. With Brad Bell's help (thank you so much!) I got on the right track to looking into PHP and realized that that was the problem. After purging the PPA and reverting to PHP 5.5.9 on the official Ubuntu repo, the problem seems to have gone away. Hopefully never to return again.

Thanks again, Brad!

EDIT: As I said below, the problem appeared again. It seems to have been solved when I disabled Craft's devMode in my environment variables. I'm not sure what would cause that to be an issue, but devMode is something I can live without for now since the site is near going live. Brad, if you have any insight as to why devMode would be causing the issues, I'd love to hear it :)

Thanks!