The 32-year-old, who was diagnosed in December after becoming one of the first British athletes to qualify for the Games, had been due to compete in the 470 class with partner Luke Patience.
"Due to Elliot’s ongoing treatment for bowel cancer, as a two person crew, they are unfortunately unable to follow the combined training programme required to maximise their chances of success in Rio," the statement said.
Selection trials for the 470 - a 4.70 metre (15.4 ft) dinghy - will continue and Patience, who won silver in London 2012 with Stuart Bithell, said he aimed to re-qualify with new partner Chris Grube.
Stephen Park, the sailing team leader, said everyone was disappointed by the deselection.
"As a team we knew we would be set a number of challenges in this journey to Rio 2016 and this is just one we have to deal with along the road," he added.
"Hopefully this doesn’t mark the end of Elliot’s Olympic journey and we look forward to seeing him back in the boat soon."
Britain is traditionally strong in Olympic sailing and has won 55 medals, including 26 golds, since the sport made its debut at the 1900 Paris Games.
Eleven British sailors have been selected for Rio so far, with five of them announced on Tuesday.
They include London 2012 silver medallist Nick Dempsey, who has been given a chance to become the most successful men's Olympic windsurfer in history if he wins gold at his fifth Games.
Double world champion Dempsey, 35, also won a bronze in Athens in 2004 and will compete in the RS:X windsurfing class.
(Reporting by Alan Baldwin, editing by Jermey Gaunt)
