In these contexts, "antic" refers to behavior that is perceived as foolish, silly, or outrageous, often done to gain attention or cause a reaction.
So, "the board won't put up with Mourinho's antics" means that the board is tired of Mourinho's silly or outrageous behavior and won't tolerate it anymore. Similarly, "I won't put up with your antics" means that the speaker won't tolerate the silly or outrageous behavior of the person they are addressing.
Overall, "antic" is used to describe behavior that is not taken seriously and is often considered inappropriate or unnecessary.