Birnstiel was born to a Swiss father and a Brazilian mother in Brazil on 1933-07-1. His family moved to Zurich, CH, when he was 5.
Between 1952 and 1959, we attended the Swiss Federal Institute of Technology in Zurich (ETH Zurich). There, he studied botany under the supervision of Albert Frey-Wyssling. His doctoral thesis, awarded in 1960, was entitled: "Über das Redoxpotential lebender und absterbender Pflanzengewebe (Tabakverbräunung)" (DOI: https://doi.org/10.3929/ethz-a-000150890).
Between 1960 and 1963, he carried out a postdoc with Professor James Bonne at the California Institute of Technology (Caltech) in Pasadena, California, US. Following this, he moved to the University of Edinburgh in the Epigenetics Research Group in the Genetics Department, having been recruited by Conrad Waddington. By the time be left Edinburgh in 1972, Birnstiel had become a professor.
In 1972, the University of Zürich offered Birnstiel a position in the university's Institute of Molecular Biology II. (The Institute of Molecular Biology I was, at this time, headed by Charles Weissmann.) Birnstiel would stay at the institute for 14 years before moving to Vienna, AT, in 1986. There, he was the founding Director of the Boehringer Institute of Molecular Pathology. He left the Institute in 1996 to return to Switzeland, where died of complications of cancer in 2014.