The prologue, showing the doomed crew of Apollo 1 banging on the hatch window, attempting to get out of the burning capsule. Then the glove goes back into the fire...
Marilyn's nightmare before the actual mission, which ended with Jim getting knocked into space.
The scene where Haise and Swaggert have started fighting. Not only are they using up more air (and producing more carbon dioxide) with all the shouting, but the prospects of a brawl in a ship whose hull is only a few millimeters thick is too horrifying to consider.
A Real Life consideration: You may thank NASA engineer John Houbolt, indirectly, for Apollo 13's survival. Early in the planning stages, NASA planned on Earth Orbit Rendezvous, using two rockets to rendezvous around Earth to form a monolithic single lunar vehicle. In short, all three astronauts would land on the moon in this one vehicle. Houbolt, after reading about the concept of Lunar Orbit Rendezvous, as developed by a Russian scientist, made several attempts petitioning NASA that EOR would make the lunar vehicle too heavy, requiring more rockets and more energy to develop. While LOR had its risks, NASA eventually agreed. They designed two independent spacecraft: One an orbiting base with an earth-return command module, and a lunar vehicle designed only to land on the moon and return. Both vehicles had independent life support, propulsion, and guidance. As it turned out, NASA realized this advantage in case of an unlikely failure and had developed Lunar Module-as-lifeboat scenario training. No one took it seriously until Apollo 13. If Earth Orbit Rendezvous were the choice, and similar problems occurred with the oxygen tanks that exploded, that single-vehicle idea would have utterly doomed this crew.