The problem boils down to how Outlook renders GIFs.
Since 2007, Outlook began to use Microsoft Word as it’s text editor.
Word cannot read HTML elements in your emails meaning that all GIFs
you include as you write an email, or GIFs received in an email from
someone else, will display as a still image of the first frame of that
GIF.
In 2013, Outlook moved to a new text editor similar to word.
Unfortunately the same HTML issues existed within this editor, so
there was no change for users.
Source
Microsoft Office FAQ
As a work-around, it is possible to conditional logic to show an animated gif to non-outlook users, and a specific static image for outlook users:
The first block of code will show an animated gif, while the second will show a static image in Outlook. Although not ideal, it does allow you to create call to actions that look effective in both environments.
Solution from Email on acid