(I originally posted this in Phrases & Sayings before seeing brawburd's succinct answer above!) A dead person is clearly no heavier than he or she was immediately before death. The phrase probably derives from the inability of the corpse to hold on to the carrier, so that the carrier can distribute the weight more evenly. But if a live person was completely immobile, then carrying him or her would seem no different to the carrier than carrying the same person dead.
I was reading something on the Internet about dead bodies. Some bloke did a load of research into the weight of the soul. When somebody died he found out that the body suddenly becomes lighter (only very slightly, 1 or 2 ounces). He did 1 or 2 experiements on Humans and a few on Dogs but his efforts we stopped due to ethical reasons. I think It was on a site called www.snopes.com
Same for a cat I would say as geofbob says for a person. When you lift a cat, you would normally yield yourselves until both comfy, distributing weight even if you don't think about it.
this is to do with muscle tension. when you lift even an unconscious person their muscles have some tension, which helps you to lift them as a whole. ie you pull on one bit and the other bits move with it. this is not the case with either corpses or aneasthetised bodies (muscle relaxants used!) as any theatre nurse will tell you. moving a corpse therefore is harder because it is more awkward- you have to be aware of the correct technique to do it easily. If you should happen to need to move a coma victim or similar out of a room (say for a fire, rather than anything illegal...) then it is best to roll them onto a rug or blanket then pull that.