The question of if it is possible to remove a chestburster before it is born while the host remains alive has been in the Alien series from the start. Throughout the movies and the expanded lore, there have been several successful chestburster extractions, although each have their own peculiarities and can be either considered non-canon or not a pure extraction.
Ripley 8 Queen Chestburster Removal
The science team of the Auriga cloned Ellen Ripley 8 times, being successful in the last attempt. Ripley 8 contained the Alien Queen embryo that had to be removed with a surgical procedure. The operation was a success and the Queen Xenomorph was removed from the host after cutting through a small umbilical cord. It can be said that this wasn't a regular case of a Xenomorph lifecycle as Ripley 8 was a clone with Alien DNA and the chestburster didn't grow as a result of a facehugging. It's worth mentioning that the Alien: Resurrection Playstation game had a portable autodock pickup - if a character got facehugged, the chestburster could be removed during a short time.
Elizabeth Shaw's Trilobite Removal
Doctor Elizabeth Shaw was not infected with a Xenomorph, but a close relative to the creature - a trilobite. The infection did not occur as a result of a facehugging, but by having sexual intercourse with a person infected with the black goo. Shaw used the autodock in the medlab of the USCSS Prometheus to remove the Trilobite creature from her stomach. Although the autodock was programmed for Mr. Weyland, it did perform a successful operation by extracting the trilobite and closing the wound. The trilobite grew rapidly and become a proto-facehugger, infecting the Engineer later. It can be speculated that the autodock would have also been successful with a Xenomorph embryo, but it perhaps would have had trouble with opening and closing the chest cavity.
Paul Church's Dead Chestburster Removal
Another self-surgery by an autodoc happened in the classic Aliens: Labyrinth comic book. In a famous flashback scene, Paul Church escaped a deformed Alien hive with a sick Alien embryo inside him. The chestburster died before being born and Church used an autodoc on his ship to remove it. After an operation lasting seven hours, the embryo was successfully removed and Church could begin his recovery. This chestburster removal is again special in the sense that the Xenomorph was dead and did not offer any resistance.
Doctor Hollis Queen Chestburster Removal
Perhaps the most "pure" chestburster removal occurs in Aliens: Defiance, a comic that was part of the "new canon" before Disney bought 20th Century Fox. Doctor Hollis was impregnated with a facehugger and carried a Queen embryo, much like Ellen Ripley. However, she was such an expert surgeon that she performed the chestburster removal herself with the help of Zula Hendricks and Davis One, while being partially sedated. The Queen was extracted together with the embryonic sack, which immediately ruptured. The chestburster tried to escape but was captured by Davis One and frozen for the time being.
Private Bella Clarison's Chestburster Removal Attempt
When Aliens: Colonial Marines was released in 2013, it forced itself into the Alien lore by resurrecting dead characters and introducing convoluted plotlines. It proposed that a chestburster is impossible to remove, contradicting other sources. In the game, the player tried to save a fellow Colonial Marine named Bella who had been recently facehugged. They reached a Weyland-Yutani surgical unit with the hope of removing the embryo. However, a Weyland-Yutani doctor revealed that the chestburster acts like cancer, infecting the whole body and killing the host even without the trauma of the birth. Moments after, Bella's chestburster was born the natural way through her chest captivity and shot in the head by O'Neill, a fellow marine.
Superman's Chestburster Digestion
Superman had a unique solution to the problem of having a chestburster inside of him in the Superman vs. Aliens comic. He was facehugged while being away from the Sun on a Kryptonian colony, and in a weakened state had to succumb to the impregnation. When he returned to Earth in an escape shuttle, his powers started to return. At the last minute before the bursting, the Man of Steel ingested the Xenomorph inside of him and barfed out a green cloud of acid. This regurgitation left him emotionally and physically wasted.
Judge Dredd's Chestburster Removal
Judge Dredd was facehugged in a hive in the undercity of Mega-City One in the Judge Dredd vs. Aliens: Incubus series. This did not hinder his progress in fighting the Xenomorphs, as he rallied other judges and used lava to melt away the Alien scourge. Soon after, his chestburster was removed by a surgical procedure in the Grand Hall of Justice with only seconds to spare. This unseen procedure seemed to be seamless, as Dredd was back on his feet soon. He went to see the creature born out of him (none of Judge Dredd's traits seemed to be inherited) and used a microwave feature of the holding cell to fry it.
Vampirella's Chestburster Birth
Vampirella is the only known being to have survived a chestburster birth. In the Aliens/Vampirella comic series, she got facehugged on a secret Mars excavation site. Soon after, the chestburster was born in a gory way, ripping through her chest. However, as Vampirella is a vampire, she had regenerative powers and started to slowly heal. She drank the blood of a nearby human that sped up her healing even more. Soon after she was fully healed and could fight the creature that was born out of her - a winged Xenomorph.
Alpha Chestburster Removal
The first Marvel Alien comic series contained two cases of successful chestburster removal. About twenty years before the main events of the story, the Colonial Marine protagonist Gabriel Cruz was cocooned in a hive and facehugged, but managed to escape with the help of a Bishop model android. Together they reached a Weyland-Yutani science lab on Epsilon station. A team of expert scientists then removed the chestburster on an operating table while it was still very small. The Xenomoprph was a special type - an Alpha that grew up to have horns and was bigger than average size. The Alpha was imprisoned on Epsilon station and there remained a special link between Cruz and the Xenomorph he birthed that came into play later in the story.
Bishop Removing a Chestburster
The second chestburster removal from Marvel's Alien was performed on Danny Cruz, the son of Gabriel Cruz. He was facehugged when the Xenomorphs escaped on Epsilon Station, an event that Danny caused together with a group of terrorists. Gabriel almost shot his son because he had little hope of removing the embryo in time. The operation was performed by Bishop, a different model than the one who saved his father years before. Although Bishop was damaged and he had not enough expertise or equipment, he successfully removed the chestburster on an escape shuttle. Gabriel Cruz fought the Alpha on the outside of the station, sacrificing himself so his son could escape.