The Furnish surname is of English origin, and seems to appear everywhere the English People traveled during the expansion of the British Empire. At this writing, there are approximately 70,000 living people with this surname in America, but it also appears in Canada, Australia, and of course Great Britain.
'Furnish' is a locational surname, meaning that it is derived from the place of origin of its bearers. In this case, the name has diverged in form. Originating in Furness, in northern England, 'Furnish' has of course diverged in spelling and pronunciation, even in Britain, where both are still found.