"tired," 1931, of unknown origin, perhaps imitative of the sound of heavy breathing from exhaustion (compare poop (n.2)). But poop, poop out were used in 1920s in aviation, of an engine, "to die." Also there is a verb poop, of ships, "to be overwhelmed by a wave from behind," often with catastrophic consequences (see poop (n.1)); hence in figurative nautical use, "to be overcome and defeated" (attested in 1920s).
It is an easy thing to "run"; the difficulty is to know when to stop. There is always the possibility of being "pooped," which simply means being overtaken by a mountain of water and crushed into the depths out of harm's way for good and all. [Ralph Stock, "The Cruise of the Dream Ship," 1921]