The UN Charter is unique in this regard because there isn't a withdrawal process established. Most documents like this have an article establishing how states can withdraw from the treaty/organization, usually by giving notice and going through a waiting period. The UN Charter has no such process. The UN's predecessor, the League of Nations, was often hampered by states threatening to withdraw if they didn't like something. The UN attempted to address many of the LoN's weaknesses, including the threat of withdrawal. (You can read the Charter here.)

The Vienna Convention on the Law of Treaties establishes that countries can withdraw from treaties based on the procedure established within the treaty (which the Charter doesn't have), or with the unanimous consent of the other countries party to the treaty (in the UN's case, the other 192 member states).

Regarding historical precedent, there is only one example. In 1965 Indonesia announced its intent to withdraw from the UN because it opposed Malaysia's appointment to the Security Council. Indonesia submitted a letter to the UN announcing its withdrawal, which the UN essentially ignored. Indonesia's flag wasn't removed from its pole outside, and its nameplate remained in the General Assembly hall. Indonesia didn't participate in any meetings, debates, votes, etc., but the UN treated the delegation as absent rather than withdrawn. The following year a coup in Indonesia resulted in a change of government, and the new government indicated that it intended to resume participation in the UN. The Indonesian delegation attended the 1966 GA session, with no re-joining process undertaken since the UN maintained Indonesia had never left.

All that said, it's difficult to predict what would happen if a country announced its intent to withdraw from the UN. If a country highly visible in the UN system announced withdrawal - one of the permanent Security Council members, for example - there would certainly be more attention given and attempts at a solution than occurred regarding Indonesia. But because withdrawal is very nearly unprecedented, we really don't know what would happen.