When I reply to a thread and click "show new replies to your posts" in the top left corner, I see a list of topics that have had replies since my last post. How do I get this to stop for certain threads?
You can't (for "Show new replies to your posts."), unless you delete the posts you made on those certain threads.
What else could it be?
The only option that you have (for such ability) is to use the "Watchlist" option instead (also in the top left corner, last option in the list).