The video is a little misleading actually because it's tough to notice exactly what happened at the end of the point. In tennis you actually are NOT allowed to hit the ball with your head - only the racquet. What happened in the video you linked to is that Federer's opponent (Juan Martin Del Potro) didn't get to Federer's backhand shot before it bounced twice - thus the point (and game) went to Federer (some people allude to this in the comments on YouTube also). The evidence backing this up is that you can hear the chair umpire announcing the game going to Federer before he even hits the ball off his head over the net. The chair umpire wouldn't announce that unless the point was already over. Since Federer had already won the game, the point was over and Federer headed the ball over the net just for fun.
Though I can't confirm it, I don't believe it is legal to hit the ball with anything other than the racquet (or paddle) in table tennis either (since I know you lose the point if the ball hits your body and not the racquet). And as far as sports that use a shuttle (like badminton), I'm not sure you could hit a shuttle with enough force using your head for it to go over the net since shuttle decelerate so rapidly once they've been hit.