Byrne was a student of Brooklyn chess coach and master John W. Collins. Collins also was known as the coach of the young Bobby Fischer, and wrote about his students in the book My Seven Chess Prodigies.
Byrne won the U.S. Open Chess Championship in 1953 in Milwaukee and around that time he achieved the second-highest rating in the US, behind Reshevsky. He was awarded the International Master title by FIDE (English: World Chess Federation) in 1962, and played for or captained five U.S. Chess Olympiad teams between 1962 and 1972. In 1972 he led a team representing Penn State University (the remainder of the team was alumni) to the US Amateur Team Championship in Philadelphia. The winning Penn State team consisted of Byrne, Dan Heisman, Steve Wexler, Bill Bickham, and Jim Joachim (alt.). Byrne's older brother, Grandmaster Robert Byrne, was also a leading player of that time.
Byrne was a great ambassador for American chess, seemingly on good terms with players from both sides of the Iron Curtain. At one point Dan Heisman (captain of the Penn State chess team under Byrne from 1969–71) said to Byrne "Gee Coach, you seem to be on good terms with every chess player in the world." Byrne paused for a moment and then said "Yes, everyone except Reshevsky" and then he laughed, as he often did, and added "But then no one gets along with Reshevsky!"
Byrne was very popular with the Penn State chess team players. Because of his illness,[clarification needed] he would walk around campus with a very wide-brimmed brown hat to keep out the sun. He would frequently tell stories about his chess exploits, often turning red from laughter. One story occurred in the 1956 Rosenwald tournament during the Game of the Century between Byrne and Bobby Fischer. Fischer was winning the game decisively and Byrne asked some of the other players if it would be a good "tip of the hat" to Fischer's superb play to let young Fischer play the game to a checkmate instead of Byrne resigning, which would normally happen between masters. When the other players agreed, Byrne played the game out until Fischer checkmated him. Byrne added "You have to remember, Bobby wasn't yet Bobby Fischer at that time", meaning that the then 13-year-old Fischer was "only" a master and not yet the 14-year-old wunderkind and top US player he would become the following year. Two other Byrne stories posted online: Fischer and the Border Patrol  and The Hustler Gets Byrned.
As a player Byrne popularized the ...a5 line in the Yugoslav Attack in the Dragon Variation of the Sicilian Defense. Against 1.d4 he often preferred to play the Gruenfeld Defense. As White he preferred using the English Opening.
|This section uses algebraic notation to describe chess moves.|
Geller–D. Byrne, Moscow 1955
1.e4 c5 2.Nf3 d6 3.d4 cxd4 4.Nxd4 Nf6 5.Nc3 g6 6.Be3 Bg7 7.f3 Nc6 8.Qd2 0-0 9.0-0-0 Be6 10.Kb1 Rc8 11.g4 Qa5 12.Nxe6 fxe6 13.Bc4 Nd8 14.Be2 Nd7 15.Bd4 Ne5 16.f4 Ndc6 17.Bxe5 dxe5 18.f5 Nd4 19.fxg6 hxg6 20.Rhf1 Rf4 21.g5 b5 22.Bd3 Rcf8 23.Qg2 b4 24.Ne2 Qc5 25.Qh3 Rf3 26.Rxf3 Rxf3 27.Qg4 Rxd3 28.Rc1 Rd1 29.c3 Rxc1+ 30.Kxc1 Nxe2+ 31.Qxe2 bxc3 32.Qg2 cxb2+ 33.Kxb2 Qb4+ 34.Kc2 a5 35.Qg4 Qc5+ 36.Kb3 Qb6+ 37.Kc3 a4 38.h4 Qd4+ 39.Kc2 Qf2+ 40.Kd3 Qxa2 41.h5 Qb3+ 42.Kd2 gxh5 0–1
- Hooper, David; Whyld, Kenneth (1987). The Oxford Companion to Chess. Oxford University Press. p. 53. ISBN 0-19-281986-0.