I keep seeing people post that if the Twitter Board rejects Elon's offer, that they will have to resign for violating their fiduciary responsibility to the company.

Who exactly is going to force them to do that?

It's about ideology and control, not profits for these people. Without a legal mechanism that would FORCE them out, I see no reason why they would leave of their own accord. In their mind they ARE doing their duty and protecting Twitter's future.