It is helpful to remember that infinitives are verbal nouns. As nouns, infinitives can be subjects of finite verbs. In English we often signal a subject infinitive with the neuter singular pronoun "it" where "it" is merely a placeholder for a following subject infinitive. "It is good to eat your broccoli." What is good? "to eat," "broccoli" being the object of the infinitive subject.
Your two examples are complicated by predicate nouns or adjectives linked to the infinitives. It helps to replace the placeholder "it" with the infinitive. The first sentence is easier:
Yet it would be your duty to bear it, if you could not avoid it.
"duty" is a predicate noun to "to bear," the second "it" being the object of the infinitive.
= Yet to bear it would be your duty, 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.
Let's break this one down:
It is weak and silly to say [main clause] = To say ... is weak and silly.
"weak and silly" are predicate adjectives to "to say"
... = <that> you cannot bear [indirect statement]
what it is your fate to be required to bear [relative clause]
"fate" is a predicate noun to "to be required"
"to bear" is an object infinitive of the participle "required."
= to be required to bear this [= what it] is your fate (to bear).
