Can all transitive verbs take to-infinitive clauses?

Yet it would be your duty to bear it, if you could not avoid it: it
is weak and silly to say you cannot bear what it is your fate to be
required to bear. — Jane Eyre

It seems ‘your fate to be required to bear’ is a to-infinitive clause (or non-finite-clause by Bas Aarts: “They would hate [Jim to sell his boat].”) and the object of cannot bear; what it is means ‘whatever it is’ and can be put in brackets.

Can all transitive verbs take the clauses as their objects?

If you see the object of cannot bear as what-clause, would you let me know just the last question?

Answer

I think there was a mistake in choosing the segment to boldface.
"Your fate to be required to bear" is not itself a constituent. It's just a part of a constituent.

(Parenthetically, the answer to the presenting question is No; only some verbs -- transitive and intransitive, because infinitives can be subjects, too -- can take infinitive complements. Of course there are other kinds of infinitives, too, but this answer is already too long.)

The smallest constituent that contains this string -- and it still needs quite a lot of untangling, because it's been done plenty of things to -- is

• what it is your fate to be required to bear

the direct object of the higher bear.

This construction is an embedded question complement clause. The introductory what gets "moved" by Question Formation from (i.e, it appears somewhere other than) its normal position in the clause, which would be after the lower bear as its direct object.
(in the dissection that follows, I mark items that are moved or deleted as code)

• it is your fate to be required to bear what

That, in turn, has been Extraposed from

• for you to be required to bear what is your fate

That is, the subject of is your fate is an infinitive clause:

• for you to be required to bear what

You is the subject of the infinitive verb be required.

Subjects are marked with for in an infinitive, and both for and you get deleted here, because infinitive subjects normally are deleted, either because they're indefinite and apply to everybody, or -- as in this case -- because they're predictable from elsewhere in the context, and you gets mentioned in your duty.

This clause, too, has been done things to. It's a Passive clause, so you has been "moved" by Passive from its position as the object of require. (What was the subject? Who knows? That's what indef means.)

• for indef to require you to bear what

with another you as the object (or requiree) of the requirement.

And we're not done yet. There's still another infinitive, the lower bear, which is intended to be in parallel with the upper bear, and to reinforce the message.

• for you to bear what

It has a subject you, in fact the same you that got "moved" by B-Raising to be the object you in the require clause above.

This whole process is what I meant in this handout by "unwinding" syntactic rules, constructions, or alternations.

Now I have to go wash my hands.