He is the son of an engineer, and his mother studied mathematics at RWTH Aachen University.
As opponents of the National Socialists (Nazis), Butzer's parents left Germany with their children in 1937 and moved to England.
During World War II, they relocated to Canada, where Butzer attended school in Montreal and studied mathematics at Loyola College (later Concordia University), completing his bachelor's degree in 1948.
In 1963, he began organizing international conferences on Approximation theory at the Oberwolfach Research Institute for Mathematics, later together with Béla Szőkefalvi-Nagy.
He studied figures such as Peter Gustav Lejeune Dirichlet, Eduard Helly, Eugène Catalan, Pafnuty Chebyshev, Charles Jean de la Vallée Poussin, the history of splines, Otto Blumenthal, mathematics in the Carolingian era, and Elwin Bruno Christoffel (on whom he published a book).