Tampons and menstrual cups do not interfere with the ring preventing pregnancy. If your ring is in when you remove your tampon or cup, you might pull it out a bit. This may be annoying if it happens a lot.
When inserting your tampon or cup, make sure that your ring is all the way in first. Then position the tampon or cup afterwards. If you do end up pulling the ring out, you can rinse it in clean water and put it back in immediately. If this happens a lot and you do not like it, but want to keep using the ring, then try using sanitary pads.
References:
- World Health Organization Department of Reproductive Health and Research and Johns Hopkins Bloomberg School of Public Health Center for Communication Programs (2018) Family Planning: A Global Handbook for Providers. Baltimore and Geneva. Retrieved from https://apps.who.int/iris/bitstream/handle/10665/260156/9780999203705-eng.pdf?sequence=1