In my experience, you can't copy and paste an animated GIF. All you get is one frame of it.
What you can do is provide the link to it. So if someone else shared such a GIF, copy the link and include it in an IMG tag.
As Steven W. said, however, movies on Twitter (and Facebook) are proprietary, so you can't even share them with a link.
IMGFLIP.com has an option for posting GIFs.
Fiddle around with it and see what you get.
Option is in the ribbon at the top on the opening page.