Shore played for his highschool hockey team the Archbishop Denis O'Connor Chargers in Ajax Ontario. On May 17, 2011, Shore committed to play NCAA hockey with the Maine Black Bears of the NCAA Men's Division IHockey East conference. While majoring in finance, Shore played 38 games for the Maine Black Bears during his freshman season. At the conclusion of the season, Shore was named to the Hockey East Academic All-Star Team and HockeyEast All-Academic Team. In his second year, Shore played 35 games as he led Maine in scoring with 14 goals and 29 assists for 43 points. Shore's outstanding play was rewarded with a selection to the 2013–14All-Hockey East First Team. He was also named a Second Team All-American and Maine Scholar-Athlete Award. On March 17, 2014, Shore was named captain of the Black Bears for the 2014–15 season. Upon completion of his junior season as Captain of the Black Bears in 2014–15 season, Shore ended his collegiate career in agreeing to a three-year entry-level contract with the Dallas Stars on March 10, 2015. He was then signed to an amateur try-out contract with the Stars AHL affiliate in Texas to finish the season. Shore was assigned by the Stars during the pre-season to begin the 2015–16 season in Texas. He got off to a quick start in his rookie campaign, as he was selected as the AHL's player of the month in October with 8 goals and 3 assists for 11 points with a plus 7 rating in 9 games. He was placed second in the AHL in goals before he received his first recall to the NHL by the Dallas Stars on November 1, 2015. He made his NHL debut against the Boston Bruins on November 12, 2015. Shore scored his first career NHL goal in a 6–5 loss against the Colorado Avalanche on October 15, 2016. In his third full season with the Stars in 2018–19, having contributed with 17 points in 42 games, Shore was traded to the Anaheim Ducks in exchange for Andrew Cogliano on January 14, 2019. Shore played out the season with the Ducks, contributing with 12 points in 34 games. On February 24, 2020, Shore was traded to the Columbus Blue Jackets in exchange for Sonny Milano.
