When a pronoun is the subject of the sentence must the pronoun be in objective case?

No, only a pronoun in the subjective case is used as the subject of a sentence.

Example: They went to the mall.

The objective case is used as the object of a verb or a preposition.

Examples:

I told them to be home at one. (object of the verb 'told')

I made lunch for them. (object of the preposition 'for')