Dance was born, the son of Eleanor (née Perks), a cook, and Walter Dance, an engineer. He was set for a career in graphic design from the Plymouth College of Art before turning to acting. He was married to Joanna from 1970 until 2004 and they have two children Oliver and Rebecca.

Dance was a member of the Royal Shakespeare Company during the mid- to late-1970s and was in many of their productions in London and Stratford upon Avon. Later he returned to the RSC to take the title role in Coriolanus at Stratford upon Avon and Newcastle in 1989, and at the Barbican Theatre in 1990. He received rave reviews and a Critics' Circle Best Actor award for his performance as the Oxford don C. S. Lewis in William Nicholson's Shadowlands, in the 2007 stage revival.

He made his screen debut in 1974, but his big break came ten years later when he played the major role of Guy Perron in The Jewel in the Crown (Granada Television, Christopher Morahan 1984), an adaptation of Paul Scott's novels that also made stars of Geraldine James and Art Malik.

He has also starred in many other British television dramas such as Murder Rooms, Rebecca, The Phantom of the Opera, Fingersmith and Bleak House (for which he received an Emmy nomination). He was name-checked in the British comedy series Absolutely Fabulous, as being slated to play the title character in The Life of Jesus Christ 2, which was filming in Morocco at the same time as the main characters of the series were there for a photo shoot.

He has appeared in numerous films, including Plenty (1985) with Meryl Streep, The Golden Child (1986) with Eddie Murphy, White Mischief with Greta Scacchi (1987), Good Morning Babylon (1987), Hidden City (1987), Pascali's Island (1988), Alien³ (1992), Kalkstein (Italy, 1992), Last Action Hero (1993), China Moon (1994), Kabloonak (1994, Paris Film Festival Award for Best Actor 1996), Century (1994), Shortcut to Paradise (Spain, 1994), Space Truckers (1996), Michael Collins (1996),What Rats Won't Do (1998), Don't Go Breaking My Heart (1998) Hilary and Jackie (1998), Gosford Park (2001), Dark Blue World (2001), Black and White (2002), Swimming Pool with Charlotte Rampling (2002), Ali G In Da House (2002), Dolls (2006) and Woody Allen's Scoop.

Dance made a cameo appearance in the 1981 James Bond film For Your Eyes Only as an evil henchman, and in 1989 he played Bond creator Ian Fleming in Anglia Television's dramatised biography, Goldeneye (the name of Fleming's estate in Jamaica and a title later used for a James Bond film).