At first I thought it might be BB Code, so I tested with other GIFs (the one above is from another website; I've tried several other types of GIFs too) and got the same thing.
The issue only seems to occur with the quick reply box and editing using the overlay. The issue doesn't occur when using the full editor. The issue also doesn't occur when you are making your own reply and adding GIFs, it's only when quoting a post with a GIF.
I disabled all add-ons but the issue remained. I've also tested here on XenForo but can't replicate the issue. I've had a brief test on Cyclechat where they use animated GIF smilies like my site does but couldn't replicate it there either.
I've also checked the browser console but cannot see any errors.
As I said, the issue only seems to occur on Chrome for Android (Chrome, FF and IE11 on Windows 7, Chrome and Safari on iPad iOS 7.1.2 were fine). The issue seems more prominent on my Nexus 7 too. When I checked with my Sony Xperia it was there but it wasn't flashing as much - the quoted text just flashed slightly (and on a few occasions there was no flashing but only part of the text displayed but was there when you looked in the BB Code editor) and the editor overlay had less flashing.
It does seem to suggest that it might be a browser issue but it must be more than that if I can't replicate it elsewhere. I want to get this fixed but I'm now at a loss as to what is causing the problem.