Tampons and menstrual cups do not interfere with the ring’s effectiveness. If your ring is in when you remove your tampon or cup, you might pull it out a bit, which might be annoying if it happens a lot.
When inserting your tampon or cup, make sure that your ring is all the way in first, and then position the tampon or cup afterwards. If you do end up pulling the ring out, you can rinse it in warm water and re-insert it right away.