Is it always necessary not to use the plural verb when "or" is used?

It really depends what exactly you are trying to say. I assume that you are talking about areas within some region of some country or continent.

I make this assumption because otherwise, it would be odd to use the definite article. Moreover, you must have in mind more than one mountainous area and more than one woodland area.

What I am trying to understand is why you choose to use the disjunction ‘or’ rather than the more obvious conjunction ‘and’.

So I have to ask you whether which of the following you are trying to state.

1. It is the case that both the mountainous and the woodland areas are amazing.

2. It is the case that either the mountainous or the woodland areas are amazing.

I have to add the possibility of there being no more than one of each.

3. Both the mountainous area and the woodland area are amazing.

4. Both the mountainous area and the woodland area are amazing.

I have added words here and there to remove any ambiguity.

All these are correct, I hope. They cannot all be what you intend the meaning of the original sentence to be. If it is 1, then your sentence is entirely unobjectionable. If it is 2, then it seems an odd thing to say, because, if the speaker had been there why would they present these as alternatives, as if one or other might not be amazing? If 3 is what is meant, the original sentence is a perverse way to say it. It could be 4.

Is it always necessary not to use the plural verb when “or” is used?

The words "always" and "never" should not be used in questions about English (or any other language.) There are no rules in English - merely guidance, and "always" and "never" create rules. Grammar is a historical record and explanation of what is current in the language - the language changes.

In your case, the answer is, and must be, "No, that is not the case."

For example,

The "either ...or" construction takes the singular with singular choices: "If you are going to buy a camera, [either] a Nikon or a Canon is a good choice."

The phrase in apposition is commonly uses "or" and a singular referent: "John, or "The Amazing Zob" as he is professionally known, is ready to show you around the circus.

The sentence "X or Y [verb]" is short for "X [verb] or Y [verb]" (something people often forget when programming, writing things like "if a or b == c" rather than "if a == c or b == c"). If both X and Y are singular, then the verb should be singular. If they're both plural, then the verb should be plural. If one is singular and one is plural, then the verb should be plural.