Enrolling at the University of Arizona, Refsnyder played college baseball for the Arizona Wildcats baseball team in the Pacific-10 Conference. In his freshman year, in the 2010 season, Refsnyder was an All-Pacific-10 Honorable Mention. After his sophomore season, Refsnyder played collegiate summer baseball for the Wareham Gatemen of the Cape Cod Baseball League. As a junior during the 2012 Wildcats season, Refsnyder batted .352, leading the Wildcats with six home runs, and recording 63 runs batted in (RBIs) and 12 stolen bases. In the first game of the 2012 College World Series (CWS), Refsnyder hit a home run. He was named the CWS Most Outstanding Player, as Refsnyder batted 10-for-21 (.476) in the 2012 CWS and the Wildcats defeated the South Carolina Gamecocks.

The New York Yankees drafted Refsnyder in the fifth round, with the 187th overall selection, in the 2012 Major League Baseball draft. Though a right fielder for Arizona, the Yankees profiled Refsnyder as a second baseman. Refsnyder signed with the Yankees on July 6, receiving a $205,900 signing bonus. He reported to the Charleston RiverDogs of the Class A South Atlantic League, where he made his professional debut on July 10. He batted .241 for Charleston.

Refsnyder met his wife, Monica (née Drake), while they were both students at the University of Arizona in 2012. She is an All-American swimmer and competed in the 2008 and 2012 United States Olympic Trials, attempting to qualify for the 2008 and 2012 Summer Olympics. They married in December 2015. Johnny Field, a teammate with the Wildcats, was a groomsman.

The Yankees gave Refsnyder an opportunity to compete for a spot with the major league team in 2015, but assigned him to Scranton/Wilkes-Barre to start the season so that he could continue to work on his defense. After he batted .290 with seven home runs, 17 doubles, and 37 RBIs in 81 games, the Yankees promoted Refsnyder to the major leagues on July 11. With his callup, Refsnyder became the fourth position player in MLB history to be born in South Korea, following Hee-seop Choi, Shin-Soo Choo, and Jung-ho Kang. He made his major league debut on July 11, and recorded his first two hits, including a two-run home run, on July 12. After playing four games, where he batted 2-for-13 (.167), the Yankees optioned Refsnyder to Scranton/Wilkes-Barre to activate Carlos Beltrán from the disabled list. He batted .225 in the second half of the RailRiders’ season.

The Yankees promoted Refsnyder to New York on September 1, 2015 as part of their September call-ups. Refsnyder played infrequently, with the Yankees initially using José Pirela, until late September, when he began to receive more regular playing time. He ended the regular season with a .302 average, and started for the Yankees in the 2015 American League Wild Card Game, which they lost.

The Yankees began to play Refsnyder as a third baseman during spring training in 2016. The team optioned him to Scranton/Wilkes-Barre on March 27. He split his playing time at third base, second base, and right field before the Yankees promoted him to the major leagues on May 17. Following injuries to first baseman Mark Teixeira and Dustin Ackley, the Yankees began to play Refsnyder at first base. He batted .262 in 122 at bats before he was optioned back to the RailRiders on August 11. He was again promoted to the major leagues as a September call-up.

After competing for a role on the Yankees’ Opening Day roster for the 2017 season, the Yankees optioned Refsnyder to Scranton/Wilkes-Barre for the start of the season. He was designated for assignment on July 19, 2017. Through July 19, Refsnyder had a .135 batting average in 37 at bats for the Yankees and a .312 batting average in 38 games for the RailRiders.

On July 23, 2017, the Yankees traded Refsnyder to the Toronto Blue Jays for Ryan McBroom. The Blue Jays assigned Refsnyder to the Buffalo Bisons of the International League, and promoted him to the major leagues on July 29, when Troy Tulowitzki was placed on the disabled list. He made his debut with the Blue Jays later that day, pinch-running in the ninth inning of a 6–5 loss to the Los Angeles Angels. Refsnyder batted .196 in 32 games for Toronto.

On November 20, 2017, the Cleveland Indians claimed Refsnyder from the Blue Jays off of waivers. He competed for a spot on Cleveland’s 2018 Opening Day roster, but batted .191 during spring training. The Indians chose Brandon Guyer over Refsnyder, and traded Refsnyder to the Tampa Bay Rays for cash considerations on March 27, 2018. Refsnyder batted ninth as the designated hitter in the Rays’ Opening Day batting order. Primarily playing against left-handed pitchers, Refsnyder batted .167 in 40 games before the Rays designated him for assignment on June 19. He elected free agency on November 2, 2018.

On November 20, 2018, Refsnyder signed a minor-league contract with the Arizona Diamondbacks, receiving a non-roster invitation to spring training. The Diamondbacks assigned him to the Reno Aces of the Class AAA Pacific Coast League to start the 2019 season. He played one game for Reno, going 0-for-4 with one strikeout, before he was traded to the Cincinnati Reds for a player to be named later on April 7. The Reds assigned him to the Louisville Bats of the International League. Refsnyder batted .315 with 10 home runs and 45 RBIs in 85 games before he was released by the Reds in August.

On December 5, 2019, Refsnyder signed a minor league contract with the Texas Rangers. He made the Rangers’ Opening Day roster in 2020. After hitting an underwhelming .200/.265/.233 for the Rangers over 34 plate appearances, including no home runs, Refsnyder was designated for assignment on August 24, 2020. He subsequently elected to become a free agent.