To disable favicons in Firefox, open the browser and type about:config into the URL bar, then press Return. You may then see a warning about how modifying advanced settings may void your warranty; click the "I'll be careful, I promise" button to proceed anyway. You'll now be looking at a long list of preferences, along with their status, type, and value. In the Filter box above the list, type browser.chrome; this will greatly shorten the list of preferences.
In the resulting list, double-click on browser.chrome.favicons to change its value from true to false. Then double-click on browser.chrome.site_icons, to set that preference to false, too. That's it; you're done. From now on, you'll no longer see favicons as you surf the net.
To reset things to normal, just reverse the process--go back into about:config and double-click the same two preferences again to set them back to true.
For Safari, first quit the browser, then open Terminal (in Applications -> Utilities), and type this command:
defaults write.com.apple.Safari WebIconDatabaseEnabled -bool NO
Relaunch Safari, and you'll be browsing favicon-free. To undo this trick in Safari, quit the browser, open Terminal again, and repeat the above command, but replace NO with YES. I've tested the solutions in both Firefox and Safari, and they work as described--though I did reverse the settings after testing, as I really do prefer to see the favicons.
Thanks to readers bdash and dangil for these tips.