Nicole writes:
Is it ever necessary to use the phrase "whether or not...?" It seems to me that the word "whether" all by itself implies both possibilities of positive or negative outcomes -- the "or not" shouldn't be necessary and indeed may be redundant. I routinely read memos in which the "Issue for Decision" is "Whether [or not] to approve a contribution of $X million to X organization."
WordzGuy suggests that in some sentences it is necessary. E.g. 
'They're going to go [to] the movies whether or not we do.'
It would be ungrammatical to say 'They're going to go to the movies whether we do.' But there are some sentences in which 
whether and 
whether or not are both acceptable.
I wonder whether they'll show up.I wonder whether or not they'll show up. (could also be worded '...whether they'll show up or not.')
Whether is similar to the complementizer 
if in such cases. They can both introduce a subordinate complement clause.
I wonder if/whether she calledI wonder if/whether she called or notThese statements with a subordinate complement clause can answer the question 
What do you wonder?But the question 
Why or in what situation do you wonder? can't be answered quite the same way. They require a subordinate modifier clause. An 
if clause is fine as an answer for both. And so it's ambiguous without context.
I'll ask if she called.It's not clear if I'm asking for information regarding her call or if I'm asking (something) on the condition that she called.
The ambiguity of the same structure using 
whether patterns differently. If I use 
whether it is unambiguously a subordinate complement clause.
I'll ask whether she called.Whether cannot introduce the modifier in that sentence.
But if we use Nicole's phrase -- 
whether or not -- we have ambiguity.
I'll ask whether or not she called.This sentence could mean
- I will request the information regarding her call
 
- If she called I will request the information. If she didn't call I will still ask for the information.
Because of this ambiguity we have an answer to Nicole's question. And it's exactly as WordzGuy suggested. There are some sentences that rely on the 
whether or not phrase. When you wish to express that one thing is true regardless of a set of possibilities you might want to use 
whether or not to introduce a subordinate modifier phrase.
In the example that Nicole provides -- 
Whether [or not] to approve a contribution of $X million to X organization
 -- the phrase is grammatical with both 
whether and with the less abridged 
whether or not. After 
whether or at the end of the sentence the 
or not tag is perfectly fine on statements that can be remodeled as a yes/no question.
He asked whether we should approve the contribution (or not).Should we approve the contribution (or not).If you're running out of breath you might want to leave it off. But there's no grammatical concern here. Interestingly -- it turns out that the constraint moves opposite to all those who suggest that language must be trimmed of all fat. There is no grammatical reason for leaving out 
or not but there is a grammatical reason for including it.
An interesting cite in the OED for 
whether. My question: is the following an example of a subordinate modifier or subordinate complement? Joseph Addison in 
The Spectator No. 92 p.5 (1711): 
Whether or no they are real Husbands or personated ones I cannot tell.