Friedrich (sometimes Frederick) Heinrich Fismer was the son of Wilhelm Fismer and his wife Friederike Charlotte, born Fitzenkotter. He studied medicine at the University of Basel, Switzerland, though he also received some training in Berlin, Prague, Vienna and Edinburgh. He served as a medical assistant during the Franco-Prussian War in 1870, and for some time as a clinical assistant in the University Hospital and University Eye Hospital in Basel. The University of Basel awarded him the degree Doctor of Medicine (MD) in 1873. His thesis was entitled Die resultate der kaltwasserbehandlung bei der acuten croupoesen pneumonie (The results of the cold water treatment of acute croupous pneumonia; Leipzig, 1873, 59p), a subject on which he had already published a paper in the Deutsche archiv fuer clinische Medicin in 1872. After graduating he came to the Cape Colony, where he was licensed to practice on 27 June 1874.
By 1884 Fismer was practising at Paarl, where he was a member of the short-lived Paarl Medical Association. This association apparently ceased to exist when its members, including Fismer, decided to join the first South African Medical Association as a body in July 1884. By 1893 he resided in Cape Town, where he was still practising in 1915. During 1894-1895 he served as a member of the Leprosy Commission of the Cape of Good Hope. Around that time he published papers on "Penetrating wounds of the eye" (1893) and "A case of cysticercus in the eye" (1895) in the South African Medical Journal (2nd series). He was then a member of the Cape of Good Hope Branch of the British Medical Association, and of the (second) South African Medical Association.
Fismer was married to Magdalene Fischer, with whom he had seven sons and three daughters.