It seems a permanent member or two often uses their veto power to dissent from the other members. Is there any mechanism for the Security Council to remove them in order to create their desired outcome? Or, in other words, what can a permanent member do to get themselves kicked off the Security Council or even out of the UN (if anything)?
The membership of the Security Council is fixed in Chapter 5 of the UN Charter:
The Republic of China, France, the Union of Soviet Socialist Republics, the United Kingdom of Great Britain and Northern Ireland, and the United States of America shall be permanent members of the Security Council.
The PRC is now a permanent member, replacing the RoC, and Russia replaces the USSR
Therefore to remove a permanent member would require an amendment to the charter, however Article 108:
Amendments to the present Charter shall come into force for all Members of the United Nations when they have been adopted by a vote of two thirds of the members of the General Assembly and ratified in accordance with their respective constitutional processes by two thirds of the Members of the United Nations, including all the permanent members of the Security Council.
Therefore, permanent members of the Security Council have a veto over their own expulsion.
However you should look to the process by which The RoC was replaced by the PRC in 1971. Essentially the process was a resolution of the General Assembly and the recognition of the PRC by the other 4 permanent members. The point of view of the other members was that the RoC was not expelled, but rather that "China" continued to be a member of the UN, and it was a matter of which delegation should be recognised. see wikipedia 1 2