Tost competed in single-seater racing during the late 1970s and early 1980s, contesting Formula Ford and Formula Three. In 1983 he won the Austrian Formula Ford Championship. Despite that success, he assessed that he lacked the ability to reach the top of the sport as a driver and changed direction, enrolling at the University of Innsbruck and the University of Vienna to study Sport Science and Management.
Following his studies, Tost joined the Walter Lechner Racing School as team manager. In 1993 he allied with driver manager Willi Weber and took charge of the WTS Formula Three team. While working within Weber's programme he encountered a young Ralf Schumacher, and Weber asked Tost to accompany Schumacher to Japan.
When Ralf Schumacher joined the Williams Formula One team in 2000, Tost followed him into the Williams orbit by joining BMW — Williams's engine supplier — as Track Operations Manager. He held that position until 1 January 2006.
In 2005 Red Bull acquired the Minardi team and renamed it Scuderia Toro Rosso. Tost was appointed team principal of the new outfit, taking charge of a small operation based in Faenza, Italy. He set an initial target of achieving fifth place in the Constructors' Championship.
Tost's best seasons with the team in terms of Constructors' Championship standing were 2008, 2019, and 2021, when the squad finished sixth in each of those years. When Red Bull rebranded Toro Rosso as Scuderia AlphaTauri at the start of the 2020 season, Tost continued as team principal without interruption.
Over his tenure the team became recognised as a development environment for young talent, with several drivers progressing from the Faenza squad to race for Red Bull Racing and other leading teams. Tost is consistently described as hard-working, professional, and direct in his communication style, unafraid to state his opinions publicly.
Following the 2007 European Grand Prix at the Nurburgring, driver Scott Speed alleged that Tost had physically assaulted him over an incident on track. Tost denied the allegation. Speed subsequently spoke to the media, describing an acrimonious atmosphere within the team and stating that both Tost and co-owner Gerhard Berger had made dishonest public statements about the matter. Speed also stated that he had no wish to race for Tost and Berger again. Speed was removed from the team's lineup during the 2007 season.
On 26 April 2023, it was announced that Tost would step down from his role as team principal at the conclusion of the 2023 Formula One season. After 18 years leading the Faenza-based outfit across both its Toro Rosso and AlphaTauri identities, he left the team at the end of that year.