At the current time, it's still possible to display the GIF animation like a video, using an .SWF container. If you make such Flash container, you can use it to display the animated GIF.
The container should be parametrized, so you don't hardcode the image link. Read the parameter in Flash, load the GIF image and display it.
When you're done with Flash, open the script link (similar to shown below) in a browser - it should display the animated image.
http://website.com/container.swf?url=http://website.com/animated.gif
Finally, fill these metatags for Facebook in your HTML:
<meta property="og:type" content="movie">
<meta property="og:video" content="http://website.com/container.swf?url=http://website.com/animated.gif" />
<meta property="og:video:secure_url" content="https://website.com/container.swf?url=http://website.com/animated.gif" />
<meta property="og:image" content="http://website.com/front.jpg"/>
<meta property="og:image:width" content="640" />
<meta property="og:image:height" content="480" />
<meta property="og:video:width" content="640" />
<meta property="og:video:height" content="480" />
<meta property="og:video:type" content="application/x-shockwave-flash" />