Try to update firefox to version 24.0 from here Mozilla.org

I've already tried updating, but I've just done it again anyway and there's been no change.

I've also tried resetting Firefox to default.

Try to boot the computer in Windows Safe mode with network support (press F8 on the boot screen) as a test to see if that helps.

Check the connection settings.

   Tools > Options > Advanced : Network : Connection > Settings
   https://support.mozilla.org/kb/Options+window+-+Advanced+panel 

If you do not need to use a proxy to connect to internet then try to select "No Proxy" if the "Use the system proxy settings" setting or one of the others doesn't work properly.

See "Firefox connection settings":

   https://support.mozilla.org/kb/Firefox+cannot+load+websites+but+other+programs+can 

Or Try Firefox Safe Mode to see if the problem goes away. Safe Mode is a troubleshooting mode, which disables most add-ons.

(If you're not using it, switch to the Default theme.)

• On Windows you can open Firefox 4.0+ in Safe Mode by holding the Shift key when you open the Firefox desktop or Start menu shortcut.
• On Mac you can open Firefox 4.0+ in Safe Mode by holding the option key while starting Firefox.
• On Linux you can open Firefox 4.0+ in Safe Mode by quitting Firefox and then going to your Terminal and running: firefox -safe-mode (you may need to specify the Firefox installation path e.g. /usr/lib/firefox)
• Or open the Help menu and click on the Restart with Add-ons Disabled... menu item while Firefox is running.

Once you get the pop-up, just select "'Start in Safe Mode"

If the issue is not present in Firefox Safe Mode, your problem is probably caused by an extension, and you need to figure out which one. Please follow the Troubleshoot extensions, themes and hardware acceleration issues to solve common Firefox problems article for that.

To exit the Firefox Safe Mode, just close Firefox and wait a few seconds before opening Firefox for normal use again.

When you figure out what's causing your issues, please let us know. It might help other users who have the same problem.

Also see here Firefox can't load websites but other browsers can

Thank you.

Issue exists in safe mode as well.

I've also been through that support page you linked to. None of that has helped. There is currently no active firewall, including Windows firewall.

Create a new profile as a test to check if your current profile is causing the problems.

See "Creating a profile":

   https://support.mozilla.org/kb/profile-manager-create-and-remove-firefox-profiles
   http://kb.mozillazine.org/Standard_diagnostic_-_Firefox#Profile_issues 

If the new profile works then you can transfer some files from an existing profile to the new profile, but be careful not to copy corrupted files.

   http://kb.mozillazine.org/Transferring_data_to_a_new_profile_-_Firefox

Created a new profile. Still nothing will load.

Also tried connecting to the internet through my phone just in case it was somehow my gateway preventing access, but it still doesn't work (and Firefox works on other computers on the same network).

Do a malware check with some malware scanning programs on the Windows computer.
Please scan with all programs because each program detects different malware.
All these programs have free versions.

Make sure that you update each program to get the latest version of their databases before doing a scan.

• Malwarebytes' Anti-Malware:
  http://www.malwarebytes.org/mbam.php
• SuperAntispyware:
  http://www.superantispyware.com/
• Microsoft Safety Scanner:
  http://www.microsoft.com/security/scanner/en-us/default.aspx
• Windows Defender: Home Page:
  http://www.microsoft.com/windows/products/winfamily/defender/default.mspx
• Spybot Search & Destroy:
  http://www.safer-networking.org/en/index.html
• AdwCleaner:
  http://www.bleepingcomputer.com/download/adwcleaner/
  http://www.softpedia.com/get/Antivirus/Removal-Tools/AdwCleaner.shtml
• Kasperky Free Security Scan:
  http://www.kaspersky.com/security-scan

You can also do a check for a rootkit infection with TDSSKiller.

• Anti-rootkit utility TDSSKiller:
  http://support.kaspersky.com/5350?el=88446

See also:

• "Spyware on Windows": http://kb.mozillazine.org/Popups_not_blocked