Favicons are cached quite heavily by browsers. So, in addition to a possible caching from a used CDN, you may need to clear the caches on the local browsers as well. Not every customer understands that if the Favicon is displayed correctly in an incognito window, a missing updated icon can only be due to the caches.
On Chrome, this cache file can easily grow up to several hundreds of megabytes, so a regular cleansing doesn’t do any harm.
Depending on the browser type, the Favicons caches can be found in different folders:
Safari
Quit Safari.
In the Finder, select Go > Go To Folder.
Enter ~/Library/Safari/Favicon Cache/ and press Return.
Use Select All (Command-A) to select the database and cached items.
Drag these to the trash or press Command-Delete.
Empty the trash.
Launch Safari.
Google Chrome
Quit Google Chrome.
In the Finder, select Go > Go To Folder.
Enter ~/Library/Application Support/Google/Chrome/Default/Favicons and press Return.
Use Select All (Command-A) to select the database and cached items.
Drag these to the trash or press Command-Delete.
Empty the trash.
Launch Google Chrome.