Dunmore played for York City, before being sold to Tottenham Hotspur for £10,500 in February 1954, which was a record transfer fee received for York at the time. Dunmore played for Tottenham Hotspur before joining West Ham United during the 1959-60 season. He made his West Ham debut on 19 March 1960 against Blackburn Rovers, and went on to make 39 appearances for the club, scoring 18 goals. He played his last game for the Irons on 4 March 1961, a home defeat against West Bromwich Albion, and joined Leyton Orient. With Malcolm Graham, Dunmore was Orient's joint top scorer in their only season in the First Division. He later rejoined York City in 1965, before moving into non-league football.