If the board president appoints a treasurer from the membership of a nonprofit corporation, after the previous treasurer had to resign because of family responsibilities, but the new treasurer causes conflicts on the board, can the board president “ fire” that treasurer if that new treasurer was never voted on by the membership in the first place?
This is ultimately a question for state law and your bylaws. It is unusual to give the president the power to remove an officer. Normally the board has the power to remove an officer of a membership organization, whether or not the membership selected the officer in the first place. And normally the board has the power to fill any vacancies in the officer ranks. These powers are not usually vested in a single individual.
In your case, if your president actually had the power to appoint the new disruptive treasurer, the president may also have the power to remove the person and then appoint a new person. But it is much more likely that both powers are vested in the board and not the president. If so, the board could remove the disruptive treasurer and replace the old treasurer with a more effective person. Check your bylaws to see what they say on removal and on replacement. If they don’t say anything, check the state nonprofit corporation law. There are usually default provisions in the statute that will apply if the bylaws don’t provide other rules.