Irish contributions total 11 of the nominations for the event, which takes place on June 11 at Radio City Music Hall, New York.
McDonagh's The Lieutenant of Inishmore is up for Best Play.
The darkly comic piece is set on a remote island, and revolves around a psychotic IRA man. And his cat.
Irish actors David Wilmot and Domhnall Gleeson and director William Milam are also nominated for their part in the play.

Shining City by Conor McPherson, about a guilt-ridden man who goes to a therapist after seeing the ghost of his wife, is nominated for two awards. The play recently opened at Broadway's Lyceum Theatre, and is up for Best Play and Best Actor, for Oliver Platt's performance.

Brian Friel's 'Faith Healer' is also in the running for several awards. The play, which debuted on Broadway in 1979, was successfully revived at the Gate Theatre in Dublin earlier this year, before traveling to New York.

Ralph Fiennes is also a contender in the Best Actor category for his performance in it, while the evil emperor from Star Wars, aka Ian McDiarmid, is another nominee from the cast of the Friel play.

John O'Fonoghue, TD and Minister For Arts, Sport and Tourism, extended his best wishes to each of the nominees.

He said: "These nominations are a testament to the vitality of modern Irish writing and stagecraft. Once again, Irish theatre practitioners are to the forefront of world audiences."

"It is particularly fitting that these nominations happen in this, the centenary year of the stage-master himself, Samuel Beckett."