I heard that James was cut from the project just because Taylor doesn’t like __________.

The correct answer is: Him

I heard that James was cut from the project just because Taylor doesn’t like him. Here, 'him' is an object pronoun used in this sentence instead of male gender. It usually comes to substitute male gender. It answer for the question 'whom' and so it is an indirect object. In general, it is categorized under object pronoun.