The 29-year-old prop or second-rower, who is captain of the French national team, suffered a ruptured bicep in Sunday's 28-8 Super League defeat at Hull.
Elima left the team's Manchester camp, where they are preparing for Friday's game at St Helens, on Tuesday morning to catch a flight back to Perpignan, where he will undergo surgery on Thursday.
Elima, who is central to France's World Cup plans and has been an ever-present in his first season back with the Dragons, is the club's third major casualty of the season.
Scrum-half Scott Dureau is still waiting to get the all-clear after undergoing surgery to remove a brain tumour, and former Leeds full-back Brent Webb has been ruled out for the rest of the season after having back surgery.
"The Olivier Elima injury is more really bad news," said coach Laurent Frayssinous.
"He is one of the leaders in the team and it's obviously a big loss for us.
"But this new long-term absence must be an opportunity for the young players who deserve to play in Super League."
Half-back Thomas Bosc, winger Damien Blanch and centre Daryl Millard all joined up with the Dragons at their Manchester base on Sunday night after missing the Hull game through injury and are expected to face Saints at Langtree Park.