Verify that you are the owner of the Firefox app directory and all files and directories under it (recursively).
If so, enter about:config in the URL bar, search for preference app.update.log, and (if not already set to true) double-click on it to change it to true. Then quit Firefox and restart. After an update check, go to Tools » Web Developer » Error Console, scroll to the end, and look for blue log messages from the update service (AUS). This should show the URL it was using to check for updates and whether there were any errors or updates.
If there are no obvious issues, copy the URL from the log and paste it into the URL bar to manually check the XML to see whether any updates are listed. The URL is generated based on app.update.url (in about:config) and includes the version you are updating from as well as platform, locale, and other information. For example this is the URL generated for Firefox 21.0 beta 6 on Mac OS X 10.8.3: https://aus3.mozilla.org/update/3/Firefox/21.0/20130430204233/Darwin_x86_64-gcc3-u-i386-x86_64/en-US/beta/Darwin%2012.3.0/default/default/update.xml?force=1. Because there is a beta 7 available it should show an update. If there are no updates for this combination of variables you should see
<updates> </updates>. If you see no updates but you think there should be updates available, check for components of the URL that do not match the example URL above (other than the specific version and build id) and try replacing them with those from the example. This will determine whether you have an unsupported product, build target, locale, channel name, os version, distribution, or distribution version.