a) Verify the page in author and also the image at the path on author? Does the file exist?
b) Verify if the image shows activated status or not on author? If yes, then check the path on publish, does the file exist? If the file exist, then check the permission of that file for anonymous user, does it have read permissions or not?
If you are accessing the image via Dispatcher , do check if gif paths are allowed in /filter rules . You can refer to dispatcher.log if its blocking the gif file.
If you are accessing the image via Publish, check if its activated or present in all publishers.