Born in Newport, Shropshire, Burton attended Adams' Grammar School. Formed at Shrewsbury Town, on 27 August 2013 he went to nearby Market Drayton Town of the Northern Premier League on a work experience loan. In September, he was loaned to Conference North team Workington, where he made one appearance, a 4–0 loss at Worcester City on 21 September. On 24 February 2014, he signed a two-year professional contract at Shrewsbury. On 30 August 2014, Burton had his first call-up for Shrewsbury, remaining an unused substitute in their 2–0 home win over Luton Town in League Two. On 23 December, he went on an emergency loan to Conference Premier side Nuneaton Town. He made his debut three days later in a 4–4 home draw with A.F.C. Telford United. His two other appearances for the Warwickshire club were clean sheets: a 1–0 win at Altrincham on 28 December for the team's first away win of the season, and a goalless draw at Telford on 1 January 2015 for which BBC Sport credited him with the result. On 17 January, Burton went to another team of the same division, Southport, on a month-long loan. He made his debut for the Merseyside club later that day, keeping a clean sheet in a goalless home draw against Chester; his three other games of his loan were all defeats. Burton made his Shrewsbury Town, and Football League debut, on the final day of the 2015–16 season on 8 May, away at Swindon Town in League One. Shrewsbury lost the game 3–0 at the County Ground, with the first goal from the penalty spot. On 9 June, he extended his contract until 2018. The following season, with Shrewsbury's two senior goalkeepers Jayson Leutwiler and Mark Halstead competing for a place in the first-team, Burton returned to former loan club Nuneaton Town on a half-season loan deal in August 2016, although this arrangement was ended prematurely on 30 September. He joined National League North side A.F.C. Telford United on an initial one-month loan deal in February 2017, which was later extended until the end of the 2016–17 season.
On 26 July 2017, Burton signed a one-year deal with newly relegated EFL Championship club Hull City for an undisclosed fee. Before making an appearance, he was loaned on 4 November to Salford City in the National League North for one month. He had his first matchday call-up for Hull on 10 March 2018 due to Allan McGregor's injury, and remained an unused substitute in the 4–3 home win over Norwich City. Later that month, he agreed to a new one-year contract. On 24 August 2018, Burton went on loan to Chesterfield of the National League until 16 January 2019. He made his debut for the Spireites the next day in a 1–0 home loss to Barnet, and was praised by manager Martin Allen after the match. He was sent off on 5 January 2019 against Ebbsfleet United also at the Proact Stadium for giving away a penalty in added time; defender Will Evans saved it to guarantee a 3–3 draw. He was released by Hull City at the end of the 2018–19 season.
Cambridge United
On 28 June 2019, Cambridge United announced that Burton had signed a two-year deal with the club.
