It's worth pointing out that, etymologically speaking, the roots of whether are which/either of two. It's inherently a "binary choice" word, so whereas "I don't know whether it be fish or fowl" is fine, "I don't know whether it be fish or fowl or good red herring" isn't really grammatical. Which is not to say people never use that extended form - but it does sometimes attract criticism.
With "unary choice" forms such as "I don't know whether I like it", the alternative ("I don't like it") can invariably be shortened to "or not" - or simply discarded completely, since it's implicit anyway.
Possibly some will say if only one choice is presented, you should use "if" rather than "whether", but skimming through written instances of "Tell me whether" suggests that most people have always been quite relaxed on that point.
TL;DR: "or not" is never required if the alternative is a simple negation of the stated proposition, but an "or" clause is required in, say, "You must choose whether to write novels or poems" (presupposing that writing, for example, software is not an option currently on offer).