Mitter was born in Schönlinde — known today as Krásná Lípa — in Czechoslovakia, but his family was expelled from the region and resettled in Leonberg near Stuttgart. He began his motorsport career on motorbikes before switching to single-seater competition in Formula Junior. He developed into the best German driver of the Formula Junior generation, recording 40 victories in the category. He also operated a commercial sideline selling two-stroke engines for Formula Junior cars. In 1963, Mitter won the Formula Junior Eifelrennen at the Nürburgring, foreshadowing a lasting and ultimately fatal connection to that circuit.
Mitter participated in seven Formula One World Championship Grands Prix, making his debut on 23 June 1963. His most significant result came at his home race, the 1963 German Grand Prix, where he scored three championship points driving a Porsche 718 dating from 1961. The performance was sufficient to attract the attention of Team Lotus, who gave him opportunities in the following seasons, though he never secured a full-time drive in the championship.
The Nürburgring's exceptional length created a practical anomaly that allowed Formula Two machinery to participate in the German Grand Prix under a separate classification, and Mitter exploited this provision in the later years of his career to maintain a presence at the event.
The greater part of Mitter's success came in endurance and hillclimb competition for Porsche, where he became a factory driver of genuine stature. He won the European Hillclimb Championship in 1966, 1967, and 1968, each time beating Ferrari's hillclimb challengers in a sustained Porsche campaign across the continent's mountain passes. His sportscar race victories included a class win at the 1966 24 Hours of Daytona in a Porsche 907. His final major victory came at the 1969 Targa Florio, where he co-drove a Porsche 908 to outright victory — a result achieved only three months before his death.
Mitter was killed on 1 August 1969 during practice for the German Grand Prix at the Nürburgring, crashing at Schwedenkreuz while driving BMW's 269 Formula Two project car. A failure of suspension or steering was suspected as the cause. The incident had immediate consequences: the BMW team withdrew both its remaining entries for Hubert Hahne and Dieter Quester, and Mitter's Porsche teammate Hans Herrmann also withdrew in solidarity. Udo Schütz, with whom Mitter had shared the winning Porsche 908 at the Targa Florio three months earlier, had survived a serious crash at the 1969 24 Hours of Le Mans and had by then already retired from racing.
Mitter's death removed one of the most accomplished German drivers of his era at the height of his powers, coming in the same year as his Targa Florio victory and continuing his contract as a Porsche factory driver.