Should countries be allowed to remain in the UN if they do not respect their citizens' rights?

The United Nations (UN) is an entity created via treaty. The member nations of the UN are signatory to the Charter of the United Nations (hereinafter ‘the Charter’), which under international law is a multilateral treaty among those member nations. Under the Charter, the member nations have a variety of obligations with respect to their internal and external politics and to maintaining the peace and safety of the world. If a member nation does not meet its obligations under the Charter, the UN will initially attempt to use it coercive power (such as sanctions) to bring the offending member nation into compliance with the Charter. The UN could also seek to enforce the Charter under international law, although international law itself is a matter of treaty and has little effect on nations that choose not to adhere to the relevant treaties. If neither coercion nor an action under international law work, the UN could expel a member nation for noncompliance with the Charter.
With respect to human rights, it is unlikely that a member country would be expelled for human rights violations. While the Charter itself affirms the member nations’ “faith in fundamental human rights,” it does not specify those rights. The Universal Declaration of Human Rights (hereinafter ‘the Declaration’), a proclamation promulgated by the UN General Assembly in 1948, does specify the human rights recognized by the UN, but the Declaration is not part of the Charter. Rather, the Declaration is aspirational in nature:

Now, Therefore THE GENERAL ASSEMBLY proclaims THIS UNIVERSAL DECLARATION OF HUMAN RIGHTS as a common standard of achievement for all peoples and all nations, to the end that every individual and every organ of society, keeping this Declaration constantly in mind, shall strive by teaching and education to promote respect for these rights and freedoms and by progressive measures, national and international, to secure their universal and effective recognition and observance, both among the peoples of Member States themselves and among the peoples of territories under their jurisdiction. 

The Declaration does not compel member states to recognize the particular rights specified therein. As the Declaration is not part of the Charter, violation of the human rights found in the Declaration would not be a treaty violation under international law and thus the UN would likely not have grounds to expel a member state for noncompliance with the Declaration. The UN could, of course, use its coercive power to try to bring member states into compliance with the Declaration, which is how the UN has frequently operated on this issue since the Declaration was promulgated in 1948.
Further, even if the UN could expel member nations for noncompliance with the Declaration, this might not be the best action. The real power of the UN is its coercive power. Expelling a member nation reduces the scope of that coercive power because the expelled nation is no longer obligated by the terms of the Charter.
https://www.un.org/en/charter-united-nations/index.html

https://www.un.org/en/universal-declaration-human-rights/