Burch turned professional with Tottenham Hotspur in July 2002 and spent a further five years at the club after signing professional forms. During that time Burch was loaned out to Conference clubs Woking, in 2003, and Stevenage Borough, in 2004, to gain some first-team experience. On 10 December 2004 he joined West Ham United on a month's loan to act as cover for Stephen Bywater whilst James Walker recovered from a groin operation; he was given the squad number 41. With Walker returning to fitness, Burch returned to Spurs at the end of the month's loan. In his time at Upton Park he was an unused substitute on six occasions. Indeed, injuries restricted Burch to just seven reserve team appearances in 2004–05. On 31 August 2005 he joined Bristol City on an initial one-month loan deal, being given the squad number 23. In his time at Ashton Gate, Burch acted as understudy to Steve Phillips whilst the club underwent a change of management with Gary Johnson succeeding Brian Tinnion. Johnson extended Burch's loan period for a second month to enable him to take a closer look at the keeper. However, Burch sustained a knee injury which saw the arrival of Adriano Basso at the club and his return to Tottenham Hotspur. He finally made his Football League debut on loan at Barnet when he was signed on an emergency 28-day loan on 30 January 2007. He made his debut for the Bees against Notts County. Burch was released from his contract by Tottenham on 19 May 2007 and signed for Sheffield Wednesday on 2 July. He made his Wednesday debut on 28 August in a 2–1 League Cup second round victory over Hartlepool United. He was released by the club at the end of the season after playing just three competitive matches. Burch agreed to sign for Lincoln City on 30 June 2008. Burch rejected a new contract with Lincoln in May 2010 and on 8 June he signed a two-year deal for League One club Notts County on a free transfer, after his contract with Lincoln City expired. In May 2012 he was released by the club, along with 12 other players.
Coaching career
On 21 June 2019, Fulham announced Burch as their new goalkeeping coach having previously held coaching roles with Tottenham Hotspur and The Football Association.
