The actor’s are said to be protesting his director’s demand that they put on a five performaces a week instead of the standard five.
It can be either of these:
The actor is said to be protesting his director’s demand that he puts on five performances a week instead of the standard five.
The actors are said to be protesting their director’s demand that they put on five performances a week instead of the standard five.
You have to make the subject and verb of a clause agree in number; that is, if the subject is plural, the verb is plural, and if the subject is singular, the verb singular.
In the first, the subject, actor, is singular, so the verb, is, should be singular. The pronoun, his, needs to be singular because it refers to the singular subject, actor. The pronoun he in the noun clause, ‘that he puts on five…’ has to be singular because this pronoun also refers to the singular antecedent, ‘actor’
In the second, the subject is plural, so each of the pronouns above need to agree/match this plural antecedent.