Most of these televised sources are heavily dubious, and need better citations.
A respiratory bypass system was a part of Gallifreyan physiology which allowed Time Lords to go without breathing for longer periods of time than most humans could. (AUDIO: Return of the Krotons) It also allowed them to filter out unpleasant smells. (PROSE: Cat's Cradle: Witch Mark)
The Doctor used it to survive more than once. The Second Doctor credited his survival on the surface of Mnemosyne to his respiratory bypass system. (PROSE: The Wheel of Ice) The Third Doctor used his respiratory bypass system to survive being strangled by the Nestene Consciousness. (TV: Spearhead from Space) The Fourth Doctor used his respiratory bypass system to survive being strangled by an Osiran service robot. (TV: Pyramids of Mars) He later used it to avoid inhaling helium. (TV: The Robots of Death) The Fifth Doctor used it to survive brief exposure to the vacuum of space, (TV: Four to Doomsday) to recover from being hanged (PROSE: A Town Called Eternity) and to retrieve the milk from a queen bat in an airless cavern. (TV: The Caves of Androzani)
The Sixth Doctor used it to avoid breathing in vorum gas. (TV: The Two Doctors) He also used it to escape hypoxia when he shut down atmospheric controls to the mines on the planet Onyakis to defeat the Krotons there. (AUDIO: Return of the Krotons) In his seventh incarnation, the Doctor used his respiratory bypass system to avoid being overcome by ether. (PROSE: Timewyrm: Genesys)
When the Doctor's second heart was extracted by Sabbath in his eighth incarnation, one of the abilities the Doctor lost was the respiratory bypass system, leaving him vulnerable to being strangled until Sabbath allowed the heart to die and the Doctor to grow a new one. (PROSE: The Adventuress of Henrietta Street, Hope, Camera Obscura)
The Tenth Doctor was able to use the bypass system to continue walking in a hospital while carrying Martha Jones, while everyone else fell unconscious. (TV: Smith and Jones) The Twelfth Doctor also used it so the clockwork droids couldn't detect him (TV: Deep Breath) and when Gus was draining the Orient Express of oxygen. (TV: Mummy on the Orient Express)
River Song had a respiratory bypass system, but she hadn't used it very much. When she and Isabella Clerkwell were trapped in a room where the air had been sucked out, Isabella passed out and River struggled to remain conscious. She tried to activate her respiratory bypass system but due to her lack of practice with it, she didn't have enough time to activate it. (AUDIO: I Went to a Marvellous Party)
The Twelfth Doctor suffered seemingly permanent blindness after he was forced to expose himself to the vacuum of space when traversing the exterior of a space station, as he was forced to donate his spacesuit's helmet to his companion Bill Potts when hers was damaged, although he survived the lack of oxygen by using his respiratory bypass system to prolong the time he spent walking in the vacuum. (TV: Oxygen)