Prior to working as a head coach, Donnan served as an assistant coach at several schools, including Kansas State University, University of Missouri, Florida State University, and the University of Oklahoma. Donnan was the offensive coordinator at Oklahoma from 1985 to 1989. From 1985 to 1988, he coached under Barry Switzer, a member of the College Football Hall of Fame and a Super Bowl winning coach. Donnan coached the legendary Oklahoma Sooner wishbone offense that helped the Sooners to three consecutive 11–1 seasons from 1985–1987, and a national championship in 1985. In 1986, the Sooners scored 508 points, which at the time was the second highest point total in the school's storied history, trailing only the Sooners 1971 NCAA record setting rushing offense that scored 534 points. Donnan was head football coach at Marshall University, where he led the Thundering Herd to a 64–21 record from 1990 to 1995, including five consecutive 11-plus win seasons, four championship games, and the 1992 NCAA Division I-AA national football championship. He was named the Division I-AA Coach of the Year twice. Donnan was then the head football coach at the University of Georgia from 1996 to 2000. He compiled a 40–19 record during his tenure. He was the first football coach in school history to lead Bulldog teams to four consecutive bowl victories. Under Donnan, the Bulldogs won the 1998 Outback Bowl, the 1998 Peach Bowl, the 2000 Outback Bowl, and the 2000 Oahu Bowl. Before the 1997 game against Mississippi State, Donnan drove a steamroller into practice and told his players they "were either going to be the steamroller or the pavement"; Georgia won the game, 47–0. Donnan was fired by University President Michael F. Adams, against the wishes of athletic director Vince Dooley, in 2000 after the Bulldogs struggled to two consecutive eight-win seasons, and three consecutive losses against Georgia Tech. Donnan's inability to return the program to the national prominence of Dooley's era and to compete with longtime SECEastern Division rivals such as Tennessee and Florida, combined with certain off-the-field problems for players, are believed to be the reasons for his dismissal. In May 2009, Donnan was elected to the College FootballHall of Fame for his coaching successes, primarily at Marshall.
Post-coaching career
After coaching, Donnan worked as a college football analyst for ESPN. In August 2014, Donnan was hired as an analyst for the American Sports Network.