Some biographers have not only disputed the common notion that Edgar

When I see something like not only... (but also), I immediately check a couple of things:
1. Whether basic parallelism is maintained
2. Whether any element that is "distributed" leads to idiomatic issues

Let's look at the 5 options:
A. Option A passes both these tests. Have not only disputed but also questioned is absolutely fine.
B. This option fails both tests. Have disputed (verb) is not parallel to over (preposition), and if we combine some biographers with the second element, we get "some biographers over whether", which is not idiomatic.
C. Can we say "have disputed whether"? Yes, we can, so this option passes the second test. It does, however, fail the first, as it asks us to make a "pure" noun parallel to something that is a little more complex. This is not the straightforward parallelism that the GMAT normally tests.
D. This option asks us to "distribute" the have from inside the not only, over to the element after but also. Some people think this is impossible. I don't, but I'm not trying to say that we should be very enthusiastic about something like this. This is a major red flag.
E. This option fails both tests. That Edgar Allan Poe drank to excess is not parallel to questioned, and if we combine some biographers have disputed the common notion with the second element, we get "some biographers have disputed the common notion questioned whether", which is not idiomatic.

So, we can safely remove B and E, and we have very strong reasons to remove C and D. We also have an option that doesn't seem to have any errors, A. At this point, I'd just mark A and move on. If we wanted to be absolutely sure, though, we could look at other issues, like meaning and tenses.

Option C leads to:
Some biographers have disputed not only the common notion that Edgar Allan Poe drank to excess but also whether he may not have drunk at all. ← Wait. So they have (a) disputed the common notion that he drank too much and (b) disputed whether he may NOT have drunk at all? That makes no sense. Not only... but also indicates a continuation. So, surely, when they say that he didn't drink too much, they would not really dispute whether he did not drink at all. That is, they may say that he did not drink at all, but they would not dispute "he did not drink at all".

Option D leads to:
Some biographers not only have disputed the common notion that Edgar Allan Poe drank to excess but also questioned whether or not he had drunk at all. ← Everyone's favorite tense, the past perfect. But are we trying to "pull" the had drunk back before something else? No, because we're looking at something that is more like a habit.

As for the whether or not, I don't have any strong opinions on that structure. I would try to avoid marking an option that contains the or not, but I wouldn't make this a "never correct on the GMAT" rule.