The more vital the worker, the more likely they will have been to be asked to postpone their annual leave in the pandemic.
Is this sentence grammatically correct? Particularly the part in bold. And would it acquire a different meaning if "to be" was dropped?
Strictly speaking, the grammar is correct. However, it's an awkward construction because of the extraposition syntax you're trying to use, as Colin's answer explains. It's not normal in speech or writing to have such a heavily conjugated "to be" verb, immediately followed by another "to be" (here as part of a passive verb).*
Here's a better way to phrase this, and still establish emphasis through extraposition:
The more vital the worker, the more likely it is that they were asked to postpone their annual leave in the pandemic.
You can play with the tense of "they were asked" however you like — use "they have been asked", "they will have been asked", etc., depending on your exact meaning.
You also asked about dropping the "to be" in your original sentence. That would not be grammatically correct, since it is part of the passive verb "to be asked".
*The essence of the problem here is that English verb conjugations rely so heavily on helper words. When a conjugated form is used together with an infinitive — in this case a passive one — there are just too many pieces to parse without effort. In cases like that, it's often better to reword the sentence to use a subordinate clause, like I suggested.
I disagree strongly with Chasly's answer.
I read
The more vital the worker, the more likely they will have been to be asked to postpone their annual leave in the pandemic.
as a heavy clause exraposition of
The more vital the worker, the more likely to be asked to postpone their annual leave in the pandemic they will have been.
and thus as both grammatical and meaningful.
Unfortunately, in either form it is awkward and confusing. In the natural form (my second one) the reader is likely to have lost track of the structure by the time they get to the main predicate "they will have been" - which is exactly why extraposition happens.
But the extraposed version has confused chasly into reading to be asked as complement of will have been, and so interpreted been to in its sense of "was meant/supposed/required to".
The more vital the worker, the more likely they will have been to be asked to postpone their annual leave in the pandemic.
This is incorrect. There is no passive verb form "to be to be asked". The following is correct:
The more vital the worker, the more likely they are to have been asked to postpone their annual leave in the pandemic.
(This assumes that the workers are still alive, or at least that this is a recent happening)
or if we really want to put the whole thing in the past
The more vital the worker, the more likely they were to have been asked to postpone their annual leave in the pandemic.
(This is a historical statement. It presupposes that we are looking back on the events after they are over)
