The first and most obvious method is based on the UN charter, Article VI, which says in its entirety:
A Member of the United Nations which has persistently violated the
Principles contained in the present Charter may be expelled from the
Organization by the General Assembly upon the recommendation of the
Obviously, if expelled from the UN, a country could no longer be a member of the Security Council.
Now, normally members of the Security Council have effective veto power, but Article 27 lists exceptions:
Decisions of the Security Council on all other matters shall be made
by an affirmative vote of nine members including the concurring votes
of the permanent members; provided that, in decisions under Chapter
VI, and under paragraph 3 of Article 52, a party to a dispute shall
abstain from voting.
The definition of disputes under Chapter VI is quite broad, so it seems likely that a country could bring a dispute to the UN focusing on a country's membership, and that country would have to abstain from voting on the issue. It could also be argued that the "recommendation of the Security Council," unlike "the affirmative vote of nine members," makes no mention of unanimity, and thus that a simple majority would suffice. Obviously this is subject to international jurisprudence, but since I doubt a case specifically dealing with this has come up, it's a viable interpretation.
Further, since this site deals with politics, not law, we must consider that in reality a country practically can be removed from the UN security council regardless of the legitimacy of the legal justifications, assuming the countries that oppose its membership and especially, the other members of the Security Council, want to remove it and are sufficiently economically or militarily powerful. If the legal methods were invalid or insufficient, they would come up with some justification to declare the nation's membership invalid in the first place.
The obvious historical precedent is Taiwan, the Republic of China, which the questioner may already be aware of. It used to be a permanent member of the UN Security Council as the successor state to the unified China that existed before 1949, until China, the People's Republic of China, became wealthy enough, and its control over the mainland obvious enough, that it made more sense for them to be a member of the Security Council. Then the PRC was recognized as the legitimate government of China, and they received Taiwan's seat on the Security Council, which they retain to this day. This was the General Assembly Resolution 2758. Not only did Taiwan lose its seat on the Security Council, but its UN membership generally. While this transition was undoubtedly made easier by Taiwan's claims to be the government of all of China, which clearly were not practically true, I think an equivalent form of reasoning could be used for any other country.
For instance, the General Assembly might decide that their membership is a priori invalid due to not being a "peace-loving state," a justification that could theoretically be applied to any country, or due to not actually being a legitimate nation at all.