Byrne rose through Irish motor racing before gaining serious attention in the Irish Formula Ford Championship in 1981. He won the Formula Ford Festival that year, the prestigious Brands Hatch invitational that regularly served as a launchpad for future stars. The victory earned him the 1982 Autosport National Racing Driver of the Year award — the first ever given — and opened the door to the British single-seater ladder.
In 1982, Byrne simultaneously pursued both Formula One and the British Formula 3 Championship — a combination that would have stretched most drivers. Despite missing a number of Formula 3 rounds to fulfil his Formula One commitments, he won the British Formula 3 championship that year. The series was then one of the strongest feeder categories in the world, and winning it while juggling top-level commitments was a remarkable display of raw ability.
His Formula One campaign came with the backmarker Theodore team, driving the Theodore TY02 with a Cosworth V8 engine. He made five entries in the 1982 World Championship, qualifying for two races — the Austrian and Caesars Palace Grands Prix — and failing to qualify for three others (the German, Swiss, and Italian). He retired from both races he started and scored no championship points. The machinery and resources available to Theodore left little margin for results; the team operated far outside the leading groups.
In October 1982, Byrne's reputation for pace earned him an invitation to test a McLaren MP4/1 at a session that included highly regarded Marlboro-backed Spirit Racing Formula Two drivers Stefan Johansson and Thierry Boutsen. According to accounts from the period, Byrne recorded a lap time faster than works McLaren drivers John Watson and Niki Lauda — despite his car being specifically detuned relative to the machinery available to the other drivers at the test. The episode became central to Byrne's story: a gifted outsider who could match anyone when given the chance, yet who never translated that demonstration into a race seat.
After a brief return to Formula Three in 1983, racing for Eddie Jordan, Byrne also contested the European Formula Three Championship that year, finishing fourth. He competed at the Macau Grand Prix and in the Formula Mondial North American Cup before shifting his direction toward the United States.
Byrne moved to the United States and began competing in the American Racing Series — the Indy Lights precursor — from 1986 with Agapiou Racing. He proved immediately competitive and over time became one of the series' most successful drivers. Across his time in American open-wheel racing from 1986 to 1992, Byrne won ten races in 55 starts, both figures ranking second in the series' all-time record books at the time. He was championship runner-up in both 1988 and 1989. He competed in the series until 1992 before retiring. Despite sustained success in America over six seasons, Byrne never made a Champ Car start — a notable omission given his record.
After retiring from competition, Byrne settled in Florida and became a driving instructor. He taught Honda Teen/Adult Defensive Driving, Advanced Defensive Driving, Acura High Performance, and Acura Advanced Performance Driving at the Mid-Ohio Sports Car Course in Lexington, Ohio. He also served as a driver coach for the Indy Lights team Brian Stewart Racing.
Byrne co-authored his autobiography with journalist Mark Hughes. Released on 8 August 2008, Crashed and Byrned: The Greatest Racing Driver You Never Saw won the William Hill Irish Sports Book of the Year for 2009. In 2016, his career became the subject of Seán Ó Cualáin's documentary Crash and Burn, which introduced his story to a wider audience.
Byrne's career is one of motorsport's most debated what-ifs. Champion in British Formula 3 in the same year he reached Formula One, a driver who outpaced Watson and Lauda in a detuned car, and a consistent race-winner for six seasons in American open-wheel racing — the evidence of his ability is substantial. That he never received a competitive Formula One seat despite the 1982 McLaren test remains the defining unanswered question of his career.