Harris, the fifth of eight children, was born in Limerick City, Ireland, the son of Ivan John and Mildred Josephine (née Harty) Harris, who owned a flour mill. He was schooled by the Jesuits at Crescent College. A talented rugby player, he was on several Munster Junior and Senior Cup teams for Crescent, and played for the well-respected Garryowen Football Club.

He might have become a provincial or international-standard rugby player, but his athletic career was cut short when he contracted tuberculosis in his teens. He remained an ardent fan of Munster provincial rugby team until his death, attending many matches, and there are numerous stories of japes at rugby matches with fellow actors and rugby fans Peter O'Toole and Richard Burton.

After recovering from the disease he moved to London, wanting to become a director. He could not find any suitable courses and enrolled in the London Academy of Music and Dramatic Art (LAMDA) to learn acting. While still a student, Harris rented the tiny "off-West End" Irving Theatre, and directed his own production of the Clifford Odets play Winter Journey (The Country Girl). The show was a critical success, but a financial failure, and Harris lost all his savings on the venture.

As a result, he ended up temporarily homeless, sleeping in a coal cellar for six weeks. After completing his studies at the Academy, Harris joined Joan Littlewood's Theatre Workshop. He began getting roles in West End theatre productions, starting with The Quare Fellow in 1956, a transfer from the Theatre Workshop.

Harris made his film debut in 1958 in the film Alive and Kicking. He had a memorable bit part in The Guns of Navarone as an Australian air force pilot who reports that blowing up the "bloody guns" of the title is impossible by air. For his role in Mutiny on the Bounty, despite being virtually unknown, he insisted on third billing, behind Trevor Howard and Marlon Brando.

His first star turn was in the 1963 film This Sporting Life, as a bitter young coal miner, Frank Machin, who becomes an acclaimed rugby league footballer. For his role as Frank Machin, Harris won the 1963 award for best actor at the Cannes Film Festival. He also won acclaim and notice for his leading role (with Charlton Heston) in Sam Peckinpah's famous "lost masterpiece" Major Dundee (1965), as an Irish immigrant turned Confederate cavalryman during the American Civil War.

He appeared as King Arthur in the film adaptation of Camelot (in which he was cast despite his limited singing range, just like Richard Burton), and proceeded to appear on stage in that role for years.

In 1966, Harris starred as Cain in John Huston's The Bible: In the Beginning. He recorded several albums, one (A Tramp Shining) included the seven-minute hit song written by Jimmy Webb, "MacArthur Park" (which Harris mispronounced as "MacArthur's Park"); that song reached #2 on the United States Billboard magazine pop chart, while topping several charts in Europe, in the summer of 1968.

A second all-Webb composed album, "The Yard Went on Forever", was released in 1969. He also wrote one of the songs, There are Too Many Saviours on My Cross, considered to be a criticism of the sectarian violence in Northern Ireland.

Some memorable performances followed, among them a role as a reluctant police informer in The Molly Maguires (1970) alongside Sean Connery. In 1971 he starred in the film Man in the Wilderness and in the low-budget Orca in 1977. Harris achieved a form of cult status for his role as mercenary tactician Rafer Janders in the 1978 film The Wild Geese. Also, in 1973, Harris wrote a highly acclaimed book of poetry, titled I, In The Membership Of My Days which was later released in record format with him reciting his poems.

By the end of the 1980s, Harris had gone a long time without a significant film role. He was familiar with the stage plays of fellow Irishman John B. Keane, and had heard that one of them, The Field, was being adapted for film by director Jim Sheridan. Sheridan was working with actor Ray McAnally on the adaptation, intending to feature McAnally in the lead role (Bull McCabe). When McAnally died suddenly during initial preparations for the film, Harris began a concerted campaign to be cast as McCabe.

This campaign eventually succeeded, and the film version of The Field (which also starred Tom Berenger) was released in 1990. Harris earned an Academy Award nomination for his portrayal. In 1992, he had a small but memorable role in Patriot Games as a member of Sinn Féin.