Burns, nicknamed "Oyster" because he sold shellfish in the off-season, was described as a "loudmouth" and having "an irritating voice and personality". Nevertheless, Burns led the Bridegrooms to an American Association championship in 1889 and a National League pennant in 1890. After retiring from baseball, Burns died on November 11, 1928 in Brooklyn, New York.
In 1883, Burns began his professional baseball career as a pitcher with Harrisburg of the minor-league Interstate Association. On the year, Burns posted an earned run average (ERA) of 2.30 over 20 games pitched, 15 of which were starts. When he wasn't pitching, Burns played second and third base.
"He was a disturber and one of the worst that ever played ball. His disposition was very bad, and he made it unpleasant for any of the boys that crested him. He is what you would call a bulldozer. [Bridegrooms manager Bill] McGunnigle may be able to handle Burns, but I doubt it."
Burns began the 1884 season playing for the Wilmington Quicksteps, but left the team after they joined the Union Association, and joined the Baltimore Orioles. Burns—the youngest player on the Orioles and the seventh youngest player in the American Association— batted .298. Despite playing in only 35 games on the season, Burns recorded a team-leading six home runs over 141 plate appearances. He continued his career with the Orioles in 1885, batting .231 with five home runs and 37 RBI, and pitching to a 7–4 win–loss record. His offensive struggles led him to be demoted to the Newark Domestics for the 1886 season, where he helped the Domestics win the Eastern League pennant. By 1887, Burns had reentered the majors for the Orioles and became the team captain until he threw a baseball at an opposing pitcher following a groundout; he was later fined $25 ($656 in 2011). On the season, he recorded nine home runs—good for third in the American Association. Burns's 19 triples were enough to tie him with five others for the league lead, and his 140 games played were tied for the league lead with teammate Blondie Purcell.
After playing in 79 games for Baltimore, Burns was transferred to the Brooklyn Bridegrooms by Harry Von der Horst, the owner of both clubs. While he was playing for the Bridegrooms, the New York Clipper described Burns as "the noisiest man that ever played on the Brooklyn team. His voice reminds one of a buzz-saw." Burns remained with the Bridegrooms for the 1889 season. He recorded team highs in on-base percentage, batting average, and home runs hit while the Bridegrooms, with an 89–48 record, became American Association champions. In the World Series, the Bridegrooms played the New York Giants of the National League. Burns hit a three-run home run to win the fourth game of the series, giving Brooklyn a 3–1 series lead. However, the Giants would take the World Series after winning five straight games.
In 1890, the Bridegrooms had moved to the National League. Burns, now 26, led the league in home runs (13) and RBI (128). He hit for the cycle on August 1, 1890—becoming the first Bridegroom to do so. The team won the National League pennant, and faced the Louisville Colonels in the 1890 World Series. The series ended in a 3–3–1 tie: bad weather led to the cancellation of more games. After the 1891 season, Burns's 1892 RBI total was third in the league, and his hits, doubles, triples, and batting average marks were the second highest on the Brooklyn team, now named the Grooms. In 1893, between games of a doubleheader, a teammate of Burns, Tom Daly, was sleeping in center field when Burns stabbed Daly with a penknife. Daly awoke and turned on the knife, leading to a severed tendon which kept Daly out for two weeks. Burns' 1894 batting average (.355) was the highest of his career; his hit and run totals were also the second highest in his career. Burns continued to play for the club until 1895, when he played for the New York Giants. In his final MLB year, Burns batted a combined .258 over 25 games.