Rees was born in Aberystwyth, Cardiganshire, Wales, the son of Doris Louise (née Smith), a shop clerk, and William John Rees, a police officer. He studied art at the Camberwell College of Arts and the Slade School of Fine Art, turning to acting when he was painting backdrops at the Wimbledon Theatre and was asked to fill a part in a play.
Rees started his career with the Royal Shakespeare Company. He played Malcolm in the acclaimed Trevor Nunn 1976 stage and 1978 television production of Macbeth. Rees created the title role in the original production of The Life and Adventures of Nicholas Nickleby, David Edgar's stage adaptation of the Charles Dickens novel, winning a Laurence Olivier Award for Actor of the Year in a New Play in 1980 and a Tony Award for Best Actor in a Play in 1982. A recorded version of the play also earned him an Emmy nomination in 1983. He also starred in the original production of The Real Thing by Tom Stoppard at the Strand Theatre in London in 1982.
Rees began to work in television during the 1970s, appearing opposite Laurence Olivier in The Ebony Tower (1984). That same year, Rees portrayed Fred Hollywell and also narrated A Christmas Carol starring George C. Scott as Scrooge. From 1988 to 1991 he starred in the late 80s/early 90s British sitcom Singles, with co-star Judy Loe. From 1989 to 1991 and in 1993, he also appeared intermittently on the long-running American TV series Cheers as the English tycoon Robin Colcord. His later television appearances include My So-Called Life as substitute teacher Mr Racine, British Ambassador Lord John Marbury on The West Wing and James MacPherson on Warehouse 13. His film career began in the 1980s. Rees played the Sheriff of Rottingham in the Mel Brooks' movie, Robin Hood: Men in Tights (1993). Rees' later film appearances include Frida (2002), The Prestige (2006) and The Pink Panther (2006).
Continuing his work in the theatre through the 1990s, both as an actor and a director, Rees was awarded an Obie Award for his 1992 performance in the Off-Broadway play The End of the Day. In 1995 he was nominated for a Tony Award for Best Actor in a Play for his role in Indiscretions. He recorded many audiobooks, including Memnoch the Devil by Anne Rice.
From November 2004 to October 2007, Rees was artistic director of the Williamstown Theatre Festival, only the fourth person to hold the post in its half century history. He replaced Nathan Lane in the role of Gomez Addams in the Broadway musical adaptation of The Addams Family, on 22 March 2010 and remained until the end of the run on 31 December 2010.
In 2012, Rees took his one-man Shakespeare show, What You Will, to London's West End, playing a three-week engagement at the Apollo Theatre.
In 2013, Rees directed Crispin Whitell's play, The Primrose Path, at the Guthrie Theater in Minneapolis.
In 2014, Rees directed Dog and Pony, a musical written by Rick Elice and Michael Patrick Walker, which had its world premiere at the Old Globe Theatre in San Diego.
His last role was as Anton Schell in the musical version of The Visit, opposite Chita Rivera, which opened on Broadway on 23 April 2015 and closed on 14 June 2015. Rees left the production in May 2015 due to his illness.
Rees was to have directed a new musical written by Elice and Will Van Dyke, Magnificent Climb, in the fall of 2016 at MCC Theater in New York. He was also scheduled to perform his one-man Shakespeare show, What You Will in New York in the autumn of 2015, and had hoped to return to the Royal Shakespeare Company for a stint in Don Quixote in 2016. He was inducted into the exclusive entertainment fraternity, the Grand Order of Water Rats as a full member.
Rees had lived in the United States for more than 25 years; he became a naturalised U.S. citizen in 2009. He converted to Judaism in the 1980s. Rees married his partner of 33 years, playwright Rick Elice, in 2011. Rees and Elice also collaborated professionally, including as co-playwrights of the comedic thriller Double Double. Elice co-wrote (with Marshall Brickman) the libretto for The Addams Family musical, the cast of which Rees had joined on 22 March 2011. In 2012, Elice and Rees received Tony Award nominations for Elice's stage adaptation and Rees' co-direction (with Alex Timbers), respectively, of Peter and the Starcatcher.
After a diagnosis of brain cancer in October 2014, Rees focused his energy on his commitment to playing opposite Chita Rivera on Broadway in The Visit, the final musical written by John Kander and Fred Ebb. While undergoing two brain surgeries, two courses of radiation and ongoing chemotherapy, Rees managed to rehearse, preview and open in The Visit on 23 April 2015. By the middle of May, it had become too difficult for him to speak, and he left the show. Rees died of brain cancer at age 71 at his home in New York on 10 July 2015. On Wednesday, 15 July 2015, the marquee lights at all the theatres on Broadway were dimmed in his honour. His ashes were sprinkled into the Atlantic Ocean. Two months later, there was a memorial service for him at Broadway's New Amsterdam Theatre.