Heinrich Klang was born in Vienna, the son of James (formerly Jacob Moses) Klang, director general of the k. k. privilegierte Versicherungs-Gesellschaft Österreichischer Phoenix, and his wife Caroline, née Rooz. From 1892 he studied law and political science at the University of Vienna and obtained his doctorate there in 1897. He worked at the Vienna Landesgericht (provincial court) until his career as a judge was interrupted in 1914 on account of the war. He served as an officer at the front and prosecutor at the military court in Galicia and then until the end of the war in the Landwehrdivisionsgericht (army divisional court) in Vienna. After the war he became a judge in the Landesgericht für Zivilrechtssachen (provincial court for civil law matters). Apart from his professional activities he habilitated in 1923 and taught at the University of Vienna. In 1925 he was awarded an extraordinary professorship and transferred to the Oberlandesgericht (higher provincial court) in Vienna. Klang was politically active in the unsuccessful Bürgerlich-demokratische Partei. He published 776 legal texts, was editor of Juristische Blätter and wrote the acclaimed Kommentar zum Allgemeinen bürgerlichen Gesetzbuch, or Klang-Kommentar.

After the annexation in 1938, Klang was dismissed as a judge and university professor because of his Jewish origins. He was forced for financial reasons to insert newspaper advertisements to sell his furniture and library of around 9,600 volumes. His hopes of a professorship in the USA and a non-quota visa were dashed because of his age, and attempts to flee to Shanghai, Cuba or Hungary also came to nothing. In 1942 he was deported to Theresienstadt, where he was employed as a guardianship judge and from 1944 as head of the ghetto court. He was also a member of the Council of Elders of the Jewish self-administration. In his autobiography he states that after the war he organized the return transport of Austrians from Theresienstadt. Klang himself arrived back in Vienna in July 1945. In November he was appointed Senatspräsident (presiding judge) of the Supreme Court of Justice and until 1946 was a member of the Constitutional Court. In 1947 he collaborated in the drafting of the Third Restitution Act and until 1949 was chairman of the Supreme Restitution Commission. He started editing Juristische Blätter again and was the first president of the Wiener Juristische Gesellschaft. Until his retirement in 1949 he worked as a judge and taught until 1951 at the University of Vienna. In 1952 he married Helene Klang, née Artner, since 1938 the divorced wife of his brother Fritz Dionys Klang, who died in 1941 at the Jewish Community hospital in Vienna's 18th district. His second brother, Marcell Klang, was deported to Mauthausen in 1942 and murdered there. Heinrich Klang, the only brother to survive the Nazi era and the Second World War, died in Vienna in 1954.

Klang had sold his law books and bookcases during the Nazi era to a lawyer, who voluntarily returned them to him in 1946. Other books went to antiques dealers in Leipzig, Berlin, Frankfurt and Vienna. In this way, some of Klang's possessions were acquired by Antiquariat Alfred Wolf and subsequently by Austrian institutions. Their origins were identified during systematic provenance research thanks to his father's bookplates. The Art Restitution Advisory Board recommended the restitution of two printed works from the Naturhistorisches Museum Wien to Klang's legal successors. In 2022, 42 books were restituted jointly by the Vienna University Library, the Library of the Vienna University of Economics and Business, the Graz University Library, the Central and Regional Library Berlin, the Badische Landesbibliothek Karlsruhe, the State and University Library Bremen, the Bavarian State Library Munich and the Saxon State Library - State and University Library Dresden.