When you go back into Minecraft having tabbed out of it, if it was in full-screen mode, the first-person hand will be gone and the paper doll will not show any animation as well as the third person player character not showing any animations.

The only animation that will show is first person held items (hitting and breaking etc.).

Some mobs will also have no animations. It seems that any mob that's loaded in when you tab out and come back in will be the mobs without animations (except for horses and perhaps a few others). - For instance I can tab out and come back in and the cow in front of me will have no animation but I can spawn one in and that spawned in cow will have animations.

Lastly - loading a different skin will give back animations to the player.