During the shortened 2012–13 season, and continuing through the 2013–14 season, the Sharks utilized Burns as a forward. Starting with the 2014–15 season, he was returned to defence. At the end of the 2015–16 season, Burns finished third in voting for the James Norris Memorial Trophy (awarded to the NHL's best defenceman); one year later, at the end of the 2016–17 season, Burns won the award for the first time in his career.
Burns was drafted by the Minnesota Wild in the first round, 20th overall, in the 2003 NHL Entry Draft. He spent most of his minor hockey career playing for the Barrie Icemen and Ajax Knights of the Ontario Minor Hockey Association (OMHA). He spent two years playing in the Metropolitan Toronto Hockey league (MTHL), now called the Greater Toronto Hockey League (GTHL), for the North York Canadiens, along with fellow NHL draftees Anthony Stewart and Geoff Platt. Burns then played his Ontario Hockey League (OHL) career as a right winger with the Brampton Battalion in the 2002–03 season. He led the team in playoff scoring that season with five goals and six assists in 11 games. Burns was runner-up for Most Improved Player in the OHL as voted on by Leagues coaches.
Upon turning professional with the Wild in 2003, Burns was converted to defence by defensively-minded Minnesota Head Coach Jacques Lemaire. Burns played in 36 games with Minnesota in the 2003–04 season, showing flashes of natural ability in rushing from the blueline. After spending the 2004–05 NHL lock-out in the American Hockey League (AHL) with the Houston Aeros, Burns adapted as a defenceman and earned a regular spot on the Wild roster for the 2005–06 season, contributing 16 points in 72 games.
In the 2006–07 season, Burns eclipsed his previous season's points total with 25 and became a significant force for the Wild in the latter months of the season, scoring back-to-back overtime winners in March and engaging in two fights during the 2007 Stanley Cup playoffs.
On October 25, 2007, during the 2007–08 season, Burns signed a four-year contract extension with the Wild. He emerged as one of the Wild's top defencemen, scoring a career-high 15 goals and 43 points.
In the 2008–09 season, Burns was regularly shifted between forward and defence with mild success before he was eventually hampered by a concussion that caused him to miss the final 19 games of the regular season. Burns' concussion was later a point of scrutiny when his agent, Ron Salcer, stipulated that Wild staff had misdiagnosed his symptoms for six weeks with sinusitis, putting Burns at an increased health risk. Burns then had shoulder surgery upon the completion of the Wild season. He finished the injury-marred season with 27 points.
Burns suffered another concussion briefly into his 2009–10 season, beginning to skate again only on January 14. He finished another injury-filled season with 20 points in 47 games.
On November 12, 2010, Burns was suspended for two games for hitting Florida Panthers forward Steve Bernier. Burns hit Bernier on the chin with the knob of the stick, causing Bernier to bleed profusely.
During the 2011 NHL Entry Draft, Burns, along with a second-round draft pick in 2012, was traded to the San Jose Sharks in exchange for Devin Setoguchi, Charlie Coyle and a first-round pick in 2011. With a year still left in his contract with the Wild, Burns signed a five-year contract extension on August 1, 2011, with the Sharks; the deal would pay him $28.8 million over five years, with an average salary cap figure of $5.76 million. In 2013, after rookie defenceman Matt Irwin's acquisition by the Sharks, in March, Burns was moved to forward, where he scored 20 points in 23 regular season games. In the 2013–14 season, Burns played the first eight games of the season, but missed 13 games after October 21 due to sore gums. On November 21, Burns returned to the Sharks' line-up and scored a goal against the Tampa Bay Lightning in a 5–1 winning effort. On November 29, against the St. Louis Blues, Burns recorded his first career hat-trick in a 6–3 victory.
In August 2014, the Sharks announced that Burns would move back to defense.
On November 23, 2016, Burns signed an eight-year $64 million contract extension with the Sharks through 2024–25, with an average annual value of $8 million per year.
Burns was named the best defenceman at the 2008 World Championships as he helped Canada to a silver medal finish as the host country. Two years later, at the 2010 World Championships, Burns led all Canadian defencemen in ice time at 18:29 minutes per game as Canada finished in seventh place. He was also part of the summer camp roster for Canada in the 2010 Winter Olympics, though he did not make the final roster cut. At the 2015 World Championships Burns was again named the best defenceman, and a member of the all-star team, as part of Canada's gold medal-winning team.
Off the ice, Burns spends his summers in Barrie, Ontario, and Lake Elmo, Minnesota, and is a noted animal enthusiast. His suburban Saint Paul home is nicknamed "Burns Zoo" due to his collection of dogs, cats and dozens of reptiles, mainly snakes. Burns Zoo was subject of a CBC Hockey Night in Canada feature with Elliotte Friedman and on After Hours.
Starting in the 2009 season, Burns purchased a suite at the Xcel Energy Center for members of the military and their families to attend Minnesota Wild home games.NHL
NHL First All Star Team: 2017
NHL Second All Star Team: 2016
NHL All-Star Game selection: 2011, 2015, 2016, 2017
NHL Foundation Player Award: 2015
James Norris Memorial Trophy: 2016-17
2008 IIHF World Championship – Best Defenceman
2015 IIHF World Championship – Best Defenceman, Media All-Star Team