The 23-year-old suffered the problem on Saturday while playing for Leinster against Zebre in the Pro12.
Moore will now miss Ireland's November Tests against South Africa, Georgia and Australia.
"He's with a specialist at the moment but it looks like he'll probably be out for around 12 weeks. It's tough on Marty," Leinster forwards coach Leo Cullen told the club's official website.
"He's been going really, really well for us, worked very, very hard over pre-season to get himself in good shape. It's going to be a big loss."
Moore has won five caps for Ireland since he made his debut for the national team earlier this year.