It's now a common trope of the genre, whether it's getting kicked off the island or being told "You're fired!" but somebody had to have come up with this first. Who created the episode-ending elimination ceremony?
The show most commonly known as the first reality competition where a contestant is eliminated in each episode is Expedition Robinson, the Swedish predecessor of Survivor, that first aired in 1997. From Wikipedia:
Expedition Robinson is a Swedish reality television program in which contestants are put into survival situations, and a voting process eliminates one person each episode until a winner is determined. The format [then called Survive!] was developed in 1994 by Charlie Parsons for a United Kingdom TV production company called Planet 24, but the Swedish debut in 1997 was the first production to actually make it to television.
After its success in Sweden (the first season finale was watched by half of the Swedish population), the US bought the format and called it Survivor. The rest is history: Survivor turned in huge ratings, became an international hit and its success and profitability pioneered the era of reality television.
[However, the genre did not have a smooth start. It's worth noting the first person who was voted off Expedition Robinson (and consequently, the first contestant ever eliminated from a reality competition) took his own life a month later.]
There is a chapter about that in Jean K Chalaby's book 'The Format Age'. The creator of the format, Charlie Parsons, was exploring new ways of doing documentaries in controlled situations. He sent 4 people to live together in very basic conditions on an island on the coast of Sri Lanka and filmed it. The project was called 'Survive!' and it was aired on the British channel ITV in 1988, but it became a flop. Later, his team added a gaming element to the format, whereby contestants voted out each other out of the game week after week. The elimination formed an essential part of the show's engine because it dictated the contestant's behavior, forming alliances and conspiring against each other. This brought tension and drama to the format. This procedure, in different variations, became a benchmark of the genre later on. The new format was rejected by every single British and American broadcaster. The Sweden broadcaster STV took a risk and produced it as 'Expedition Robinson', that later became 'Survivor' on the American television.