Born in Brechin, Scotland, Orr began his musical life playing an organ built by his father, an amateur cabinet-maker. He studied at the Royal College of Music and Cambridge before traveling to Italy to study with Alfredo Casella and to Paris to work with Nadia Boulanger.
After returning to England, Orr taught at two schools and at Leeds University. In 1938, he was appointed organist at St. John's College Cambridge. After serving in World War II, he taught at Cambridge and the Royal College of Music.
In 1956, he was named chair of the music department at Glasgow University. Six years later, the founders of Scottish Opera asked him to become its first chairman. He served in the post until 1976, overseeing the company's move into the Theatre Royal. Meanwhile, he was appointed professor of music at Cambridge in 1965, the first Scot ever to serve in the post.
His compositions include the song cycle From the Book of Philip Sparrow, premiered by the Scottish National Orchestra; the operas Hermiston and Full Circle, performed by the Scottish Opera; and the Rhapsody for Strings and Symphony in One Movement.