Hey Paul, it's our understanding that Facebook does not support manually uploading an animated GIF as a "photo". So, if are you using the Facebook share option right from within Snagit or saving the GIF and then manually publishing to Facebook, they will only display a fixed picture.
The method I found to be supported is posting a web URL for the animated GIF. To do this with Snagit, you can share to Screencast.com or Dropbox first. However, there is one more step to get the full URL to the GIF file itself.
Open Snagit->Preferences->Outputs: - Screencast.com -> Choose "Full URL" from the "Clipboard Contents" dropdown menu - Dropbox -> Uncheck "Shorten share link"
Now share to either of those to get the full URL to the GIF.
Hopefully that all made sense :), if you have any questions please let me know!