Let’s say I become a citizen of a country that doesn’t allow dual citizenship. During naturalization, new country B tells me I have to renounce citizenship from old country A.
Does that have any effects back in country A? How would country A know? Would country A even care if they found out?
Yes, renouncing your citizenship is a formal process.
Not all countries require you to actually go through the process, some just have it in the oath.