I might be able to answer my own question here.
Edit: Looks like it doesn't play unless I tap on it to open. Did I do something wrong or is that just a feature of Lemmy?
Depends on what client you use. They play for me in Summit.
It autoplays for me in multiple browser UIs (lemmy-ui, Photon, Tesseract, Alexandrite) as well as in the Summit, Voyager, and Thunder mobile apps. Though the frame rate is ridiculously fast in Thunder for some reason.
That's a sexy robot you have there, fella.
Works on voyager. Also syncs well with rasputin.
Working on Jeroba
Reporting animated gif sighting in Boost.
Works on Sync
Works on Boost with "animations play by default" on.
Depends I think on how you submit the link.
The preview from old.lemmy.world plays as expected.
Only in the thumbnail for Alexandrite. it shows as a link when you click the post and opens in a new tab/window. It would work if the image were submitted with  like below.
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