Animated GIF images show the animation if they are hosted on a different site. This one is hosted on Photobucket.
When you want to insert a image from outside your own site make sure that you click the "Web address (URL)" option and then insert the image address. If the address is correct a preview of the image should appear.
This one is on 4shared.com
This came from Google+
This is in a Dropbox folder.
And this one in in Google Docs (Google Drive).
Upload the GIF to Google Docs & share it, then right click on the image and select 'View Image Info'. From the list of URLs select the relevant Image (the preview should be animated) and copy the URL. In Google Sites, edit the page and Insert...Image, selecting the Web address option and paste in the copied URL.
Address such as
Must be changed to: