Furrer was born in Wörgl, Austria (then part of Germany). After the end of WW2 his father was expelled from Austria, and the family found a new home in Kempten, Bavaria. He stayed there till he joined the University of Kiel to study physics. He later changed to the Free University of Berlin, where he received diploma in 1969, and doctorate in 1972. In 1974 he became assistant professor in Stuttgart, in 1979 qualified for full professorship. 1980-1981 he spent at the University of Chicago, 1981 at the Argonne National Laboratory in Chicago, USA.
In 1977 Furrer applied in the astronaut selection for the first spacelab mission and made it into the final selection, however Ulf Merbold was finally selected. In 1982 the astronauts for the first German spacelab mission were selected from the finalists of the first selection, and Furrer was one of the two selected. He then was payload specialist on STS-61-A (D1), which was launched October 30, 1985. The other payload specialists were Ernst Messerschmid and Wubbo Ockels (Netherlands).
After his spaceflight he became professor in 1987 and also director of the institute of space sciences at the Free University of Berlin.
Furrer was an avid pilot. He made his pilot license in 1974, doing several long distance trips with one engine planes - like a flight over the inland ice of Greenland in 1979, or a solo flight from Germany to Quito, Ecuador in 1981. His love for planes did finally cost him his life, as he died in a plane crash during a flight show in Johannisthal (Berlin). He was passenger on a flight with a historic Me 108, after the end of the official flight show, when the pilot Gerd Kahdemann lost control of the plane and it crashed shortly after 6pm. Both the pilot and Furrer were dead immidiatly.