Theissen graduated from RWTH Aachen University with a diploma in mechanical engineering and joined BMW in 1977, working in the engine calculation department. Over the following years he took on progressively senior responsibilities within BMW's engine development division. In 1989 he earned a doctorate in engineering from Ruhr University in Bochum. In 1991 he became head of Product Concepts at BMW, and in 1992 was appointed Director of Advanced Drivetrain Development. He was promoted to Director of BMW Technik GmbH in 1994. In 1998, in addition to his BMW Technik responsibilities, he established the BMW Technology Office in Palo Alto, California.
In April 1999 Theissen was appointed BMW Motorsport Director alongside Gerhard Berger, taking responsibility for the company's Formula One programme and its other motorsport activities, including factory entries in the World Touring Car Championship and at the 24 Hours of Le Mans.
BMW had a history in Formula One stretching back to its turbo engine supply to Brabham in the 1980s, during which Nelson Piquet won the 1983 World Drivers' Championship. In the contemporary era BMW entered a partnership with Williams Racing in 1998, and by 2001 BMW was credited with having the most powerful engine on the Formula One grid. Despite a strong championship challenge in 2003, the relationship between BMW and Williams deteriorated publicly, with Theissen criticising the team's inability to build a package capable of winning the constructors' championship and Williams countering that the engine was insufficient. This friction contributed to BMW's decision to purchase the Sauber team.
Theissen became team principal of BMW Sauber when the team entered Formula One in 2005. In their first season the team finished fifth in the constructors' championship, both a significant improvement on Sauber's previous standing and a match for Williams' position in the same year. In 2007 BMW Sauber finished second in the World Constructors' Championship after McLaren-Mercedes were excluded from the standings following a spying episode.
The high point of Theissen's tenure came at the 2008 Canadian Grand Prix, when Robert Kubica led Nick Heidfeld home in a one-two finish, giving BMW its first victory as a Formula One constructor. When BMW announced its decision to sell the team back to Peter Sauber and withdraw from Formula One, Theissen elected to leave the sport alongside the programme, though he retained his position as BMW Motorsport Director.
Theissen retired from BMW on 30 June 2011, succeeded by Jens Marquardt. His tenure at BMW Motorsport represented a period in which the manufacturer pursued Formula One both as an engine supplier and, eventually, as a constructor in its own right.