Moriz Grünebaum was the son of Gustav Grünebaum, k. k. Hofrat and head of the Bauabteilung der Staatsbahnen (building department of the state railways), who was made a hereditary peer in 1876. After studying law in Vienna and Kraków (doctorate in 1905), Moriz entered the service of the Niederösterreichische Statthalterei (Lower Austria governor's office). In 1910 he became librarian of the k. k. statistische Zentralkommission (Imperial Royal Main Statistics Commission), transferred in 1925 to the University of Vienna library and in 1931 to the Akademie der bildenden Künste in Wien (Academy of Fine Arts in Vienna), from where he retired four years later as state librarian first class. He lived at Liechtensteinstraße 45A in Vienna’s 9th district with his wife Laura (née Pollak, after leaving the Jewish Community in 1910 Pernier, 15 August 1869 –22 October 1940), owner of the lingerie business John & Pernier, and his widowed mother Charlotte Grünebaum (née Forchheimer, 27 February 1849–14 October 1941). Grünebaum, who had been a member of the Österreichische Exlibris-Gesellschaft since it was founded in 1903, collected prints, ex libris and books and also had an extensive collection of "Mandlbögen" (Biedermeier cutout sheets for paper theatres) from Kunstverlag Trentsensky in Vienna. He published regularly in the Österreichisches Jahrbuch für Exlibris und Gebrauchsgraphik and in newsletters by the Österreichische Exlibris-Gesellschaft and himself illustrated one of the two fairy tale books that his mother published during the First World War. He had ex libris made by Felix Hochstimm and Hans Przibram.

After the annexation of Austria in 1938, Grünebaum was considered a Jew, in spite of his conversion to Catholicism in 1895. The Institut für Denkmalpflege (Institute for Monument Preservation) was particularly interested in disposing of his "Mandlbögen" and entered into negotiations with the Niederdonau-Sammlungen (Lower Danube Collections), the present-day Landesmuseum Niederösterreich (Lower Austria Provincial Museum) in that regard. Grünebaum had to store his art collection with the transporter Spedition J. Z. Dworak junior when he moved in October 1940 after Laura's death with his mother to sister Margarete Fürth, widow of the doctor and chemist Otto Fürth. They lived only a few months in Döbling before Moriz, Margarete and their daughter Wilhelmine were forced to move into a "Sammelwohnung" (collective apartment) in Herminengasse. In August 1942 Moriz Grünebaum was deported to Theresienstadt, where he perished at the end of the year at the age of sixty-nine. Margarete and Wilhelmine Fürth had been murdered shortly before in Maly Trostinec.

Between 1948 and 1957, works from Grünebaum's turned up on the Vienna art market. The auction house S. Kende sold over 1,500 Mandlbögen in several lots to the present-day Wien Museum; in 1957, the Albertina acquired a drawing from the art dealer Leopoldine Zelenka; and in 1986/87 three prints that had also been sold in 1953 by S. Kende were part of a major donation to the Academy of Fine Arts in Vienna. Because of his stamp, the inscription "Fuer Pflicht und Ehr", the initials MG and an emblem, the objects were clearly identified as having come from Grünebaum's collection. It is no longer possible to reconstruct when and how the works came onto the art market. In February 2017 the Vienna Restitution Commission determined that the Mandlbögen in the Wien Museum should be restituted. In its decision of 11 January 2019, the Art Restitution Advisory Board recommended the restitution of the sheets in the Academy and the drawing in the Albertina.