The sentence contains an example of the common omission of that after certain verbs, particularly in informal English. For example:
He said (that) he was tired.
The government required (that) everyone be vaccinated.
Restoring the missing that to the quoted sentence gives us:
Chekhov's gun is a dramatic principle that requires that every element
in a narrative be necessary and irreplaceable, and that everything
else be removed.
So now it is clear that the second that in the original introduces a parallel structure following requires, with both that-clauses being dependent on principle.
If the sentence were to be consistent in its that-omission, it could also read:
Chekhov's gun is a dramatic principle that requires every element in a
narrative be necessary and irreplaceable, and everything else be
removed.
This lack of consistency is probably what causes the confusion.
In response to OP's request in the comment for the grammar of 'verb + that + noun + bare infinitive', see Wikipedia's entry on the present subjunctive, including this extract:
The main use of the English present subjunctive, called the mandative
or jussive subjunctive, occurs in that clauses (declarative content
clauses; the word that is sometimes omitted in informal and
conversational usage) expressing a circumstance which is desired,
demanded, recommended, necessary, or similar. Such a clause may be
dependent on verbs like insist, suggest, demand, prefer, adjectives
like necessary, desirable, or nouns like recommendation, necessity; it
may be part of the expression in order that... (or some formal uses of
so that...); it may also stand independently as the subject of a
clause or as a predicative expression.
