The game simply doesn't support leaving factions without user-made events and/or decisions. Joining a faction is kind of like marriage, except the only way to leave is through death.
To remove a nation from a faction in your save, here's what you do. First make a backup copy of the save and put it somewhere else, in case you break something by accident. Then, open the save in your favorite text editor. I use Wordpad, but anything will work.
Once you have it open (might take a while), search for "faction={", which will take you down to the top of the diplomacy section. There, find the faction you want and delete the entire line for the nation(s) you want to remove. The file in ../HOI3/common called countries.txt has a list of what nation each 3-letter tag translates to.
If the nation you want to kick out is at war with someone and you want to end the war, then you'll need to do another search, this time for "active_war". Scroll down the list to find the war you're looking for, and delete the line containing the nation's tag you want to remove from the conflict.
From there, save the file and restart your game to see if it worked. If it didn't, delete the file you edited, copy the backup and put it in the save games folder, and try again. Make sure you keep your backup save safe. Be aware that after you kick a nation out of a faction, if the nation is AI controlled it will likely rejoin within a few days if it can.
a lot.
