William Furness spent his youth in Philadelphia, where his father, a native of Boston, was pastor of the Unitarian church. On finishing school, he was started in a "counting room," but after a year told his father that he was unsuited to business. His father encouraged him to pursue his preference instead, and he turned immediately to art-apparently without any formal training. He worked first in Philadelphia, then was briefly in Brooklyn. He was settled in Boston by 1850.
Initially Furness worked only in crayon, enjoying high favor for his portraits of distinguished citizens of Boston and Philadelphia in this medium. The critic Henry Tuckerman noted that "in the estimation of many he stood next to Seth Cheney in this line." In 1852 he went to Düsseldorf to study painting with Emanuel Leutze; he also visited Dresden, Munich, and Venice and may have stayed in Europe up to four years. From the time he returned to America until late in 1863 or early 1864, Furness seems to have stayed in Philadelphia. By spring 1864 he had rooms in the Boston Studio Building, where he came to know Elihu Vedder. It is likely that he resided mainly in the Boston area but spent much time in Philadelphia during his few remaining years.
Furness first exhibited in an Academy annual in 1860; his name was incorrectly spelled in the exhibition catalogue as Furnace. He was represented in each annual thereafter through 1865 and, finally in 1867, the year of his death, with two portraits sadly noted as unfinished. He also exhibited at the Pennsylvania Academy of the Fine Arts and the Artists Fund Society, both in Philadelphia; the Boston Athenaeum; and the Washington (D.C.) Art Association. A number of Furness's oil portraits were shown in the Philadelphia Centennial Exposition of 1876.