Here are the sentences in question:
- (original) A prominent question in metaphilosophy is that of whether philosophical progress occurs, and more so, whether such progress in philosophy is even possible.
- (dropped 'of') A prominent question in metaphilosophy is that whether philosophical progress occurs, and more so, whether such progress in philosophy is even possible.
Here are a couple of major changes that dropping "of" introduces:
The word that in #1 forms part of the phrase that of, which refers back to the opening phrase. It (that) may be replaced by the phrase the prominent question while retaining the meaning of the sentence. In #2, it acts as a conjunction, introducing the following clause. That and the prominent question aren't 'drop-in replacements' for each other in #2.
The phrase whether philosophical progress occurs is the immediate "prominent question" in #1. In #2, it is a dangling clause. Removing of to produce #2 causes the whole sentence to become incomplete. Instead of the question being "whether X is Y" as it is in both whether causes of #2, it needs to be a statement along the lines of whether [condition], [conclusion], asserting that regardless of the truth of the condition, the conclusion holds.
Is the “of” necessary in “that of whether”?
Since you want to retain the meaning of the sentence you quoted, yes, it is necessary. However, if you remove "that of" instead of just "of", you retain the meaning of the original sentence, as noted by Janus Bahs Jacquet in comments to this answer.