Burgess began his career at Arsenal's academy and after he was released he went on to play with the youth and reserves teams at non-League Bishop's Stortford before deciding to study history at the University of Birmingham. Whilst in his second year university, Burgess earned a trial at Middlesbrough through a coach of the university's first team.
Middlesbrough
He impressed at Boro and he signed a two-year contract in July 2012.[2] He made his professional debut on 4 May 2013 in a 2–0 defeat away at Sheffield Wednesday.[3]
Hartlepool United
He joined Hartlepool United on loan in August 2013.[4] The loan deal was extended until the end of the season in December.[5]
Burgess scored his first goal in professional football in a 5–0 win for Hartlepool in a Football League Trophy game against Bradford City in September 2013.[6]
Peterborough United
On 19 August 2014, Burgess signed on a month's loan for Peterborough United.[7]
He completed a single game on loan, a 2–1 home defeat by Sheffield United, before completing a permanent transfer on 21 August 2014. Burgess signed a four-year contract for an undisclosed fee.[8][9]
Portsmouth
On 25 June 2015, Burgess signed for south coast club Portsmouth as part of the summer overhaul at the club by new manager Paul Cook.
He went on to make his debut on 11 July 2015, in a pre-season friendly against local Conference South club Havant & Waterlooville.
During the 2015–16 season, in which Portsmouth reached the play-off semi-finals, Burgess made 43 appearances in all competitions, partnered in most games by either Matthew Clarke or Adam Webster.
2016–17 season
Following a successful first campaign for Pompey, during which he established himself as the club's first-choice centre back, Burgess began the 2016–17 season as one of the team's central figures. In total, Burgess made 45 appearances and scored four goals over the course of the season, as Portsmouth won the League Two title. Burgess, along with teammate and fellow defender Enda Stevens, was named in the League Two team of the year; a reward for their part in Portsmouth's league-best defensive record.[10]
2019–20 season
Burgess was awarded the EFL League One Player of the Month award in January 2020 and won the award for Portsmouth Player of the Season in the same year.
Union SG
Burgess turned down a contract extension at Portsmouth and on 7 July 2020 joined Belgian First Division B club Union SG, signing a three-year deal with the option of an additional year.[11] He scored his first goal for the club in a 3–1 victory against Club NXT. At the end of the season USG were promoted, making it the second league title victory of Burgess' career. Burgess was awarded a place in the Team of the Season at the end of the season.[12]
On 21 November 2021, Burgess scored two goals for the club in a 7–1 victory against Oostende.[13] The team finished first in the regular season of their return campaign, although they would lose out on the title during the six-game playoffs, finishing second.[14]
The following year, RUSG once again qualified for the championship playoffs, having finished level on points with Genk.[15] In a close fight for first on the final day, Burgess and Union would lose out when they conceded three late goals against Club Brugge on the final day, thus losing out to Antwerp by a point.[16] During 2023, Burgess, who had missed just three games throughout the league season, was given the nickname "Air Burgey" as a nod to his aerial prowess.[17]