That is just an indicator of whether there are new posts to the thread since you last viewed it. It cannot be disabled, but it also does not affect your Tracked Thread list, does not trigger email notifications, does not let others know you viewed it, etc. Basically, just ignore it for threads you're not interested in. But if you want to follow a thread, click the Track this Thread button.
Hope this helps,
If I click on a thread out of curiosity and then decide it isn't something I'm interested in, the red "New Post" alert still appears for it every time someone adds to it.
Is there a way to turn that off for threads you're not interested in?