But what if there's no saving him?
Just to clarify: nobody technically "dies" on a plane. For legal reasons, the flight crew can't declare a person dead, and the airlines take serious issue with doctors on board doing so as well. Blame the lawyers for that one. But if the person really is dead, the flight must go on, and it's up to the crew to determine where to put the body.
Chances are, corpses will remain in their seats until the plane lands
The FAA actually has no set rules on what to do with in-flight corpses, so procedures vary from airline to airline. And since it's not a common occurrence -- unlike, say, a fistfight breaking out over a seat-reclining incident -- most FAs don't know the procedure off hand. In fact, the most common response we got from FAs and pilots was, "I’d have to check in the manual."
That said, people who have died mid-flight in the past have simply been left in their seats until the plane lands, at which point medical personnel removes the body. Typically, the body is covered with a blanket so passengers aren't staring at a corpse all flight, although some airlines do carry body bags on board.
Other times, the body has been moved up to first or business class, since those seats offer more room and are often empty. Alas, how our friend on British Airways ended up with a deceased seatmate.
If there's absolutely no room on board and the decision is made to move the body from its original seat, the flight crew has to get resourceful. Dead bodies are specifically not to be put in the lavatory, but they can be placed nearby.
"A lady just died on my friend's flight,” said one International FA. "They left her on the floor between the bathrooms, tied up a few blankets, and told everyone the lavatories were inoperable."