Burns was born and raised in Bath, Somerset. He attended Beechen Cliff School where he first started playing rugby union, moving on to Hartpury College to continue his studies, and being picked up by Gloucester to play for their academy side. He made his first team debut for Gloucester on February 2012, coming off the bench in a 15 - 23 loss to London Irish in the LV Cup. That year, he would also have success for the England Under 18 side, winning the European Championships held in Spain, scoring 7 points in the final against Ireland which England won 25 - 13. During the 2012-13 season Burns was sent out on loan to Hartpury College to gain experience of competitive men's rugby. He made 17 appearances in National League 2 South, scoring 71 points which included 6 tries, helping the club to 3rd place in the division. He also made several appearances for Gloucester in the LV Cup and was part of the sevens side that won the 2013 Premiership Rugby Sevens Seriesin the summer, scoring a conversion in the final victory against Leicester Tigers. The 2013-14 season saw Burns build on his sevens success with Gloucester, signing a new contract and making debut appearances in the Heineken Cup and Aviva Premiership, but most of the season was spent back on loan at Hartpury College. He had a very successful second loan spell, helping Hartpury to claim the 2013–14 National League 2 South league title and promotion to National League 1, and being one of the league's top points scorers with 205 points. He was also called up by the England Under-20 team for the Six Nations. In the summer he played a key role in England Under-20s victory at the 2014 IRB Junior World Championship held in New Zealand, scoring 11 points as England triumphed 21 - 20 over South Africa in the final. The following season Burns got more game time at Gloucester, accumulating 17 appearances across all competitions and making the bench for the side that won the 2014–15 European Rugby Challenge Cup. In 2015-16 he made appearances at fly-half and full-back, before sustaining a knee injury against Wasps in March 2016, which ended his season. He was Gloucester's first choice fly-half for the 2016-17 and 2017-18 seasons, but after the club signed Danny Cipriani, he asked to be released from the last year of his contract, and joined Ulster on a two-year contract in July 2018. In November 2019, having established himself in the Ulster team and formed a strong half-back parternership with John Cooney, he agreed a renewed contract until June 2022. Burns is Irish qualified by virtue of a paternal grandparent. He was called up to an Ireland training camp by new head coach Andy Farrell in December 2019, and was named in Ireland's 2020 Six Nations Championship squad.
