Here is a situation from a SGA (Student Government Association). My colleauge and friend Chris Jachimowiicz from Muhlenberg University gave this answer and I had to share.
Situation- Our Speaker of the House for our SGA is proposing three 'resolutions' at our meeting this week. What is the formal process of temporarily removing him as chair and appointing someone else as chair for the duration of debate over each of his proposed resolutions?
Answer- For those wishing to replace the chair for this meeting, it will be important that they caucus ahead of time to be sure that they have enough votes (two-thirds) to carry the action out effectively. If this is merely the attempt of a minority, it will fail.Your first step should be to attempt to reason with the chair, perhaps in advance of the meeting, and to explain that s/he should step down from the (impartial) chair if s/he intends to make motions or engage in debate on motions. It is wholly permissible (and strongly suggested) that a chair "pass the gavel" to a vice-chair or other member in order to have the position of chair be seen as impartial in the conduct of the meeting. Failing that, understand that members DO have the right to install a chair pro tem and use the process below.
1. When the meeting starts, stand and address the chair thusly: "Mr Chair, I rise to a Question of Privilege" (this does not require a second). The chair should acknowledge you and respond "state your question".
2. State: "I move that we suspend the rules and appoint [member] to serve as Chairman Pro Tem for the duration of this meeting [or through the duration of new business]."
3. The chair should seek a second on the motion. If a second is secured (any member may second without recognition by the chair to speak), the motion should be voted on immediately as it is not debatable. It requires a vote of two-thirds for passage.
4. If it is successful, the member or officer named in the motion should assume the chair and continue the meeting. The previous chair, if a member of the organization, should take a seat with the assembly.
A. It is possible that the chairperson will rule the motion to suspend the rules out of order on the grounds that the governing documents grant the right to chair to his/her office. The argument would continue that a change to the governing document would require an amendment.
A1. If the chair rules the motion "Out of Order", a member should immediately rise to "Appeal from the Decision of the Chair" (you do not need to be recognized by the chair). The motion requires a second and is debatable in a limited fashion - the chair MAY debate, being provided two times to speak (open and close). All members are entitled to speak once.
A1a. Given your time to speak, you should explain the following to the membership:
A "Motion to Suspend the Rules" requests that particular "Rules of Order" be set aside for a limited period of time. In RONR, Rules of Order "relate to the orderly transaction of business in meetings and to the duties of officers in that connection." [RONR 10th ed. p. 15, lines 5-7]. As such, who chairs a meeting is a Rule of Order and can be suspended EVEN IF that rule of order appears in the Constitution or Bylaws [RONR 10th ed. p. 17, lines 19-21].
A2. Upon completion of the debate on the "Appeal from the Decision of the Chair," the question put before the assembly should be, "Shall the decision of the chair be sustained?" Those in favor of removing the chair should vote AGAINST the motion; they do not want the decision of the chair to be upheld. A majority AGAINST the motion would lead to its desired defeat.
A3. If the Appeal is defeated, then the question of Suspending the Rules should be allowed.
B. It is possible that, even after the chair has lost the Appeal, s/he refuses to put the question of suspending the rules to the assembly.
B1. It is at this point that a member, standing at his or her seat, may again offer the motion to the members without the chair's assistance. An ally should second the motion, and then the member making the motion should conduct the vote (probably to the loud protestations of the chair). Again, a vote of two-thirds is required.
B2. If the motion passes (keep in mind that a vote of two thirds always requires either a show of hands or a rising vote), then the individual recently appointed as chair pro tem should leave his/her seat and stand in front of the assembly, perhaps with the deposed chair at his/her back. The chair pro tem would then conduct the meeting.
C. If the deposed chair becomes unruly, the chair pro tem should attempt to call the member (e.g. chair) to order. If the chair remains unruly, the membership may (must) vote to have the unruly member removed from the hall - the chair alone may not order this action.