I love a good GIF - you only have to look at my Twitter - and I'd even say that using them to illustrate my posts and for commentary (with mostly pop-culture and film orientated clips) is a big part of my personal visual brand. Copyright and regional legislation is a confusing nightmare on GIF usage, so here's my 'official' (what I tell clients and panicky account managers) and 'all in one place' take on the matter.
Fair Usage
It all depends on how you define 'fair usage'. Put simply, this means that if the original content is used for a "transformative" and limited reason, like a parody, for commentary or criticism (again, see my Twitter) it's ok under US and UK law.
Transformative means that they are using the source work in a totally new or 'unexpected' way - like remixes by Pogo or CassetteBoy. This is because they don't undermine the market for the original work. Basically, people aren't going to watch a Mad Men or Parks and Rec. GIF instead of watching the original programme.
Fair use law creates an opening for copyrighted material to be remixed and repurposed, as long as the new use doesn't create any economic competition for the people or organisation that holds the actual copyright - yes, you can make money out of sound bites etc., within reason.
What are your intentions?
Where it gets tricky is the notion of 'intellectual property' and intent. If you own footage it's (obviously) cool to make GIFs out of it, but if you don't (like, say, making GIFs out of footage of sporting events and tweeting them out saying "Goal!") that's naughty. Why? Because the original content is made to show just that - goals - and pulling out the highlights devalues the original. During the Olympics, the International Olympic Committee had a zero tolerance, no-GIF policy, for this very reason. The owner of the footage would be perfectly within their rights to slap you with a cease and desist order and sue your plagiarist ass.
Another area is individual representation. By this, I mean that performers and celebs have the right to use local "right of publicity" laws that let public figures control how their image is used. This is usually applied to defamatory actions and mash-ups, however, rather than just clips - though how the clip is associated (for example, politically) could be a trigger for this. Don't take people out of context or repurpose them (too much).
On the whole, if you want to be sure, get written permission from the copyright holder and any actors, embed GIF content that's not our own (and let the site you link to take the heat - thanks Giphy) or make your own GIFs from our original content. Avoid sports, especially recent stuff. I'm an individual, not a brand, my personal accounts are totally a different matter and I'm comfy with that - do as I say (to be sure) not as I do. Use those that are inherent to platforms - use those provided IN Twitter, Facebook etc. then (while not ideal) it's down to the platform.
Clear as mud? In short, yes, you're probably ok. Legally it's a grey area, just don't use them in a defamatory way or for financial gain and don't take the piss. If you're a business/brand jump through some hoops.
Enjoy.