There is a common story that there are nonprofit boards and their CEOs who don't get along. Not getting along is a real possibility although the reality of this tension is frequently not anywhere near what goes on in the workplace or family. The primary reason: "boards" as an entity don't tend to have anywhere near the frequency of contact with the CEO as do family or workplace members. Based on conversations, aside from the board chair, many nonprofit board members spend as little as 15 hours a year doing their job. Yes - that's nine meetings a year with an average of one-and-a-half hours a meeting.
So, how on earth do boards even have time to have a conflict? Well, stuff happens - sometimes there are reports that don't make sense; financial irregularities or just plain trouble; sometimes employee complaints; sometimes community members; sometimes a board meeting (well 2-3-4 board meetings) are so unsatisfactory in terms of substance and even pleasantness, that board members get irritated and turn their irritation back onto the CEO - justifiably so in my opinion.
And yes there are longer term solutions like a post board meeting exec session where the chair is tasked to go chat with the CEO or an annual goal setting and subsequent performance review and a governance committee that structures with the chair a positive engagement where the board does get to exercise it's fiduciary duty.
For sure, nonprofit boards don't HAVE to have a negative relationship with their CEO. CEOs in-turn might want to value their board members and help them do their job. But sometimes, it's just not going to work for either party. Action will be necessary and in that case, yes, it's up to the board to resolve - they are after all the surrogate (for the public) "owner" of the nonprofit - and let's not forget this.