How can I pause animated GIFs in Messages.app on OS X?
You can not pause an animated GIF.

But build-in in your operating system is the power of App Nap which is developed just for your needs. Shamelessly copied from the 'Power Efficiency in OSX' PDF from apple.com. It is an easy read.


  App Nap
  
  While your Mac can run many different apps at the same time, the
  reality is that you’re typically interacting with only one of them,
  and in some cases (such as using a full-screen app), you can’t even
  see what the other apps are doing. App Nap takes advantage of this
  fact by putting applications that you can’t see into a special low-
  power state that regulates their CPU usage as well as network and disk
  I/O. App Nap can be automatically triggered in the following
  situations: 
  
  • The app’s windows are not visible.
  
  • The app is not playing audio.
  
  • The developer has not explicitly made the app exempt from App Nap by using the existing IOKit IOPMAssertion API (currently used in OS X to prevent the system from sleeping while an app is busy).
  
  • An app is in background and hasn’t drawn recently.


Btw: GIF is pronounced as pronounced as ɡɪf or dʒɪf ;)