Conor Michael Sheary was born June 8, 1992, in Winchester, Massachusetts to Kevin and Robin Sheary. He grew up in Melrose, Massachusetts with his two sisters, Caitlin and Courtney. Sheary dated Jordan Sullivan since their years at Cushing Academy in 2008. They became engaged in March 2016 and were married August 2018.

On July 1, 2015, Sheary was signed to a two-year entry-level contract with the NHL affiliate of Wilkes-Barre, the Pittsburgh Penguins.

In the following 2015–16 season, Sheary was recalled from the AHL to the Pittsburgh Penguins on December 15, 2015 and made his NHL debut on December 16, 2015 in a 3–0 loss against the Boston Bruins. On December 18, 2015, he scored his first NHL goal on goaltender Tuukka Rask of the Boston Bruins and registered his first NHL assist in the same game.

On March 13, 2016, Sheary recorded his first multi-goal game, scoring 2 goals against the New York Rangers at Madison Square Garden against goaltender Henrik Lundqvist. He would then record his first career playoff goal vs Lundqvist on April 21, 2016. On June 1, 2016, Sheary scored the game-winner, in overtime, to win against the San Jose Sharks 2–1 in the Stanley Cup Finals. On June 12, 2016, Sheary won the Stanley Cup with the Pittsburgh Penguins in a 3–1 victory against the San Jose Sharks.

In the 2016–17 NHL season, Sheary continued to play alongside Sidney Crosby for the entire year, getting 23 goals and 30 assists for 53 points. Sheary got his first 3-point game on January 16, 2017, against the Washington Capitals. The Penguins won the game 8–7 in overtime, with Sheary scoring the overtime winner. The Pittsburgh Penguins later went on to win the Stanley Cup, giving Sheary his second Stanley Cup ring.

On July 30, 2017, the Penguins re-signed Sheary to a three-year, $9 million contract worth $3 million annually.

In an interview in November 2017, Sheary admitted that commentators, fans, and players have been mispronouncing his name. His name is pronounced SHARE-ee not SHEER-ee.

On June 27, 2018, Sheary, along with teammate Matt Hunwick, were traded to the Buffalo Sabres for a conditional fourth-round pick in the 2019 NHL Entry Draft. In doing so, the Penguins cleared over $5 million in salary cap space. In the 2018–19 season, Sheary played in a top-nine forward role, contributing with 14 goals and 34 points through 78 games as the Sabres missed the post-season.

At the NHL trade deadline Sheary was dealt by the Sabres, along with Evan Rodrigues, and was re-acquired by the Pittsburgh Penguins in exchange for Dominik Kahun on February 24, 2020.