Furneaux was born at Swilly near Plymouth. He entered the Royal Navy, and was employed on the French and African coasts and in the West Indies during the latter part of the Seven Years' War (1760–1763). He served as second lieutenant of HMS Dolphin under Captain Samuel Wallis on the latter's voyage round the globe (August 1766 – May 1768). Furneaux was made a commander in November 1771, and commanded HMS Adventure which accompanied James Cook (in Resolution) on his second voyage.
On this expedition Furneaux was twice separated from his leader (February 8, 1773 to May 19, 1773; and October 22, 1773 to October 14, 1774, the date of his return to England). On the former occasion he explored a great part of the south and east coasts of Tasmania, and made the earliest British chart of the same. Most of his names here survive; Cook, visiting this shore-line on his third voyage, confirmed Furneaux's account and delineation of it (with certain minor criticisms and emendations), and named after him the Furneaux Group of islands in Banks Strait, opening into Bass Strait, and the group now known as the Low Archipelago.
After Adventure was finally separated from Resolution off New Zealand in October 1773, Furneaux returned home alone, bringing with him Omai of Ulaietea(Raiatea). This first South Sea Islander to travel to the United Kingdom returned to Tahiti with Cook in 1776–1777.