Raised in Winnetka, Illinois, on the North Shore of Chicago, Byrum attended New Trier High School, and later studied at New York University in the late 1960s. His instructors included Haig P. Manoogian and graduate student Martin Scorsese, and classmates Oliver Stone and Eric Jenkins. At NYU he co-wrote "Item 72-D, The Adventures of Spa and Fon" with Director Edward Summer.

Byrum directed his first feature film Inserts with stars Richard Dreyfuss, Jessica Harper, Veronica Cartwright, Stephen Davies, and Bob Hoskins. Byrum followed soon after as writer/director of Heart Beat starring Nick Nolte, John Heard, and Sissy Spacek, as Neal Cassady, Jack Kerouac, and Carolyn Cassady. It was on this film that Byrum started his long collaborative relationship and friendship with Composer Jack Nitzsche. Byrum teamed with "Mahogany" Producer Rob Cohen which resulted "Scandalous" and "The Razor's Edge", the 1984 film directed by Byrum based on the W. Somerset Maugham novel starring Bill Murray in his first dramatic role as "Larry Darrell", co-starring Catherine Hicks, Theresa Russell, Denholm Elliott, and James Keach. Byrum and Murray co-wrote the screenplay. The film faltered with critics and audiences, as most were not ready to accept Murray in a non-comedic role. In later years the film achieved cult status and many fans have traveled to its international locations, to replicate Larry's spiritual journey.