As I understand it, if you reply to a thread, any new replies will make the thread show up in the list of topics when you click "Show new replies to your posts". The only way to keep that from happening is to remove any posts you've made to the thread. You'll see people post "following" on threads so the thread always shows up under "Show new replies" for them.
If you subscribe to a thread by clicking "notify", you will get an email for new replies. However, you can unsubscribe to that clicking "Unnotify".