A group of directors of our nonprofit organization publishes a letter to the community and signs it "The Board” of our organization with a partial list of board members. They use the organization's name and logo to give a sense that they are speaking with an authoritative voice. That's a false representation if the list of people identified in the letter as "the board" is not the full board. Correct? But what is the legal offense? For those directors who were excluded from the decision to write or publish the letter, do they have recourse?
I assume that the subset of directors involved in this activity is a minority of the board and that they have no authorization from the full board to do what you are objecting to. (If they are a majority, they can simply authorize the activity and there would be nothing to complain about. In that case, those who have been excluded would have to consider whether they want to stay around).
If the group of publishers is an unauthorized minority and talking to them directly about the situation doesn’t get you where you want to get, you could ask the majority of the board to approve a specific policy prohibiting such activity. Violation of the policy would then give a specific ground for disciplinary action, which, depending on your state law and/or your bylaws, might give you a basis for removing some or all of them from the board. If removal by the board fails and you have a voting membership, you might be able to encourage the members to remove them, or at least not re-elect them at the end of their terms.
If the removal or threatened removal route does not resolve the issue, those who are dissatisfied could send a communication to “the community” saying that these missives do not speak for them and should be not viewed as the official position of the board. It would unfortunately escalate an internal difference into a public dispute that could undercut the organization’s reputation and its program. But that might be appropriate if your directors can’t act like adults and work it out.