Greer's father was Daniel David Milne Willard, a test pilot at the United States Naval Test Pilot School. His mother, Suzanne, daughter of the late Navy test pilot Seymour Anderson Johnson and Alice Virginia Kelley, was a homemaker. In 1965, after his mother remarried, to Vice Admiral Howard E. Greer, D.M.W. Greer began a lengthy period of 20 different schools in 17 cities, before graduating from Radford High School in Honolulu, Hawaii in 1975.
Greer subsequently began his university education at the United States Naval Academy (class of 1980). Transferring to Oregon State University, Greer earned a bachelor's degree in architecture/landscape, and on the same day, he was commissioned an ensign in the U.S. Navy. Earning his wings in 1982 at Pensacola, Florida, he reported to an anti-submarine warfare helicopter squadron in San Diego, California, where he flew the Kaman SH-2 Seasprite.
In 1983, Greer was transferred to the Naval Information Office on Fifth Avenue in New York City where he was responsible for selling the Navy story to the major media outlets and assisted in the planning of the Statue of Liberty Centennial celebrations. In 1986, Greer left the U.S. Navy to pursue other interests.
Greer worked as a freelance graphic designer for various magazines and illustrated travel books, for a time, working for a multi-media production company. He also studied acting, appearing in several off-off Broadway productions in New York and other small roles in before turning to writing for the theatre.
In 1992, Greer wrote his first play, Burning Blue, based on his knowledge of events surrounding the treatment of gay servicemen in the U.S. Navy, which premiered at The King's Head Theatre on London’s fringe in 1995. The play opened to great critical acclaim. West End producer Robert Fox took it to The Theatre Royal, Haymarket, where the play received Evening Standard nominations and won two Olivier Awards: for John Napier’s set design and David Hersey’s sound design. The play opened in Cape Town and then Johannesburg in 1996.
Burning Blue then opened in Tel Aviv’s Beit Lessin Theatre, where it also won critical praise and played to full houses for 18 months. In 1998, John Hickok returned to direct the American premiere in Los Angeles at the tiny Court Theatre, where it was an instant hit with the Los Angeles Times. Kevin Otto (from the South Africa production) reprised his role as Lieutenant Dan Lynch, and Martin McDougall – who originated the role of Lieutenant JG Charlie Trumbo in London – returned to play the role again. They were joined by Tim DeKay, who impressed the critics with his portrayal of Lieutenant Will Stephensen, originally played by Irish actor/director Ian FitzGibbon in the London. Productions followed in San Francisco later that year and finally in New York in 2002.
The film adaptation, starring Trent Ford and Rob Mayes and directed by Greer, began production in 2010. Released in 2013 as a feature film, it met with negative critical reviews.
Best friends Lieutenants Daniel Lynch and Will Stephensen are U.S. Navy fighter pilots flying the McDonnell Douglas F/A-18 Hornet, hoping to become the youngest pilots to be accepted into the space program. After two accidents, one of which is due to Will’s failing eyesight, their unit is subject to an NCIS investigation led by John Cokely.
At the same time, a third pilot, Matt Blackwood, arrives on the carrier and quickly develops a close friendship with Dan, driving a wedge between Dan and Will. Cokely’s investigation leads to him uncovering rumours about Dan and Matt’s relationship just as they both begin to fall in love. When Matt decides to leave his wife and move in with Dan, there is a third accident and Cokely’s investigation ramps up the pressure on Dan.
Initial reviews for Burning Blue were predominantly positive, but many American critics took a more contrary view as later staging did not reflect the changing acceptance of homosexual relationships. Don Shewey in The Advocate wrote, "DMW Greer’s play Burning Blue takes a Reefer Madness approach to homosexuality. Ostensibly a fictional condemnation of the U.S. military’s hypocritical "don’t ask, don’t tell" approach to gay servicemen, this far-fetched melodrama presents homosexuality as a kind of science-fiction virus transmitted by the heady combination of red lights and disco music." He wondered, "And how on earth has such a nonsensical sub-soap-opera script gotten productions in London, Los Angeles, San Francisco, and New York?"
Dan Bacalzo in a review of a New York performance, questioned why, "... 'Burning Blue' is supposedly set after President Clinton's 'Don't Ask, Don't Tell' policy was instated, yet there is no specific mention of that policy." His review ended with the opinion that, "At its best, 'Burning Blue' is a vehement rebuttal to a military practice that is unfair and discriminatory. But unlike Marc Wolf's Another American: Asking and Telling, which also addressed the topic of gays in the military, Greer's play overstates its case without ever achieving the emotional depth that the topic merits."
Greer's second play, Alice Virginia, premiered in London in 2003, with Andrew Halliday and Amanda Boxer as the secondary leads and Susannah York playing the title role.