Moody and former Cardiff City manager Malky Mackay are under investigation by the Football Association after a dossier was reportedly sent by Cardiff owner Vincent Tan detailing the alleged misconduct.
The Daily Mail reported that Moody and Mackay allegedly sent racist, sexist and homophobic texts to one another during their time together at the Welsh club.
A statement on Palace's official website said: "In light of the events of yesterday, sporting director Iain Moody has tendered his resignation and it has been accepted with immediate effect. There will be no further comment from the club on this matter."
Mackay was believed to be close to being appointed as Palace's new manager following the departure of Tony Pulis last week, but following the emergence of the allegations, talks broke down.