Dunn was born in County Durham, being brought up in the village of East Howle and attending school in Durham. At the age of 12, Dunn was involved with the youth team of Burnley for four years before joining Derby County.
Dunn made his debut with hometown club Bishop Auckland in 1971, before joining Scarborough in 1975, and soon endearing himself to their supporters. With Scarborough he gained two FA Trophy winners medals, scoring in the 1977 final at Wembley against Dagenham. He made 265 appearances for Scarborough in all competitions, scoring 65 goals.
He is sometimes confused with another long-standing Scarborough player called Harry Dunn, who made over 900 appearances for the club and was already at the club before Dunn joined from Bishop Auckland. To avoid confusion, the new signing was given the fictitious middle initial 'A' to differentiate between the two players. Therefore, to Scarborough fans he has always been known as Harry A. Dunn. It was the original Harry Dunn who managed Scarborough in 1982, and between 1984 and 1986, and was clubman of the year in 1970 and 1974, and not Harry A. Dunn as it is often reported.
He joined Blyth Spartans in 1981, winning a Northern League Championship medal before rejoining Bishop Auckland in 1983.
He was manager of Bishop Auckland until 1993 when he went to Fleetwood Town with Phil Staley. In 1995, he left to manage Blyth Spartans. However, he did not stay long at Croft Park as he was appointed as manager of Whitby Town in December 1995, staying in charge until his sacking in September 2004. He returned to Blyth by October 2004.
On 8 November 2008, his Blyth Spartans side defeated Shrewsbury Town of Football League Two, in the first round of the FA Cup. Blyth then went on to defeat A.F.C. Bournemouth 1–0 in a second round replay at Croft Park before losing 0–1 to Premier League club Blackburn Rovers on the same ground in round three, a game which Dunn refused to be moved to Newcastle United's St James' Park ground. Alongside his managerial duties for Blyth, Dunn worked in a caretaking capacity for a Durham nursing home. On 27 April 2009, Blyth Spartans confirmed that the club would not be renewing Dunn's contract.
Dunn was re-appointed as manager of Whitby Town in May 2009 and resigned on 4 October 2010.
He was appointed as chief scout at Darlington 1883 in June 2012.