Heiberg studied abroad and, after a period as Norwegian consul in China, returned to Norway, where he financed the creation of the Ringnes brewery in 1876 together with the brothers Amund Ringnes (brewer) and Ellef Ringnes (administrator and salesman).
Together with the shipowner Thomas Fearnley, the brewery sponsored the polar expeditions of Fridtjof Nansen and Otto Sverdrup, and funded the construction of the exploration vessel Fram. This led to Heiberg's name being given to Axel Heiberg Island in Canada, the Axel Heiberg Glacier in Antarctica, and the Geiberg Islands in Siberia.
In 1878 Heiberg was one of the founders of the rowing club Christiania RK. Later he founded the "Consul Axel Heiberg and Manufacturer Hans B. Fasmer Fund" (in 1915 transferred to the Fridtjof Nansen Fund). In 1898 he was one of the founders of the Norwegian Forestry Society, and was chairman until 1923. He also funded the statues of Henrik Ibsen and Bjørnstjerne Bjørnson outside the National Theater in Oslo.