Marr was born in Durham, England, the son of Lieutenant Colonel John Lynn Marr, a director of two shipbuilding firms and of the Sunderland Forge and Engineering Company. When his father died in 1931, ten-year-old Leslie inherited the baronetcy originally held by his grandfather, shipbuilder Sir James Marr, 1st Baronet, though he chose not to use the title for most of his life. He was educated at Shrewsbury School and Pembroke College, Cambridge, where he studied engineering, graduating in 1942. During World War II he served as a technician in the Royal Air Force, and it was during a posting to Palestine that his interest in painting first developed.
After the war, Marr studied under the influential painter David Bomberg at what was then the Borough Polytechnic, now London South Bank University. He became closely associated with Bomberg's circle of students, known as the Borough Group, and allocated the upper floor of a rented bookshop as an exhibition space for the group. Following the group's dissolution in 1950, Marr continued to paint and travel across Britain and continental Europe. Between 1983 and 1991 he lived and worked on the Isle of Arran, later settling in Norfolk. His works are held in public collections including the British Academy, Imperial College, the Laing Art Gallery in Newcastle upon Tyne, and Pallant House Gallery in Chichester.
It was during his early 1950s travels that Marr took up motor racing. He competed in two Formula One World Championship Grands Prix, both in a privately entered Connaught. His debut came at the 1954 British Grand Prix on 17 July 1954, where he finished thirteenth. At the 1955 British Grand Prix he retired after a damaged brake pipe caused him to spin off the circuit.
Outside the World Championship, Marr achieved more satisfying results. He won the 1955 Cornwall Motor Racing Club Formula 1 Race and finished fourth at the 1956 New Zealand Grand Prix, demonstrating that his commitment to racing, though that of an amateur, was far from casual.
Marr married three times. His first wife was artist Dinora Mendelson, daughter of London art dealer Jacob Mendelson; Mendelson's mother Lilian Holt later became Marr's former teacher David Bomberg's second wife, creating an unusual connection between his artistic and personal lives. Marr and his first wife divorced in 1956. He subsequently married Lynn Moynihan in 1962, and later Maureen Monk. He died in Gimingham, Norfolk, in May 2021 at the age of ninety-eight. The baronetcy passed to his first cousin twice removed, Allan James William Marr.