Bryan Garner, Garner's Modern American Usage, second edition (2003) answers this question in a consistent way that is easy to apply to particular instances of "whether (or not)":
Whether or not. Despite the superstition to the contrary, the words or not are usually superfluous, since whether implies or not—e.g.,:
- "In another essay, 'The Rules of the Game,' he discusses moral codes and whether or not {read whether} they work." Diane Hartman, "At Life's End Carl Sagan Awed by Life's Unknowns," Denver Post, 22 June 1997, at D6. [Other examples—involving "deciding whether (or not)" and "decide whether (or not)"—omitted.]
But the or not is necessary when whether or not means"regardless of whether" {the meeting will go on whether or not it rains}. [Other examples omitted.]
By Garner's reasoning, you don't need or not in this sentence:
You can decide whether to go once you've finished your homework.
but you do need it in your original sentence:
You can go whether or not you finished your homework.