One of the things MRI do to create images is to pulse magnetic fields during imaging. These pulses are far weaker than the main field but will cause vibration of metals due to Lenz's and Faraday's laws. As the magnetic field changes it induces current in the ring, current in the ring interacts with the magnetic field to produce a (small) force on the ring. If you were getting a head/brain scan in a typical MRI your hand will lie near the locations within the scanner that see the largest swings in magnetic field. Beyond the vibration, rings are generally too small to be a heating issue even at 3T.
Best practice for at least a decade is to always remove all rings and all jewelry and failure to detect rings or other jewelry is generally seen to indicate a problem in screening. That is... if a radiologist sees evidence of a ring on the images there better be an explanation. The reality is that particularly older people have not removed their rings in decades and their joint disease may have expanded so much that it simply cannot be removed and the risk/benefit doesn't justify damaging the ring nor denying them the benefits of a scan. But if the patient can't take the ring off, the magnet wont either.
Just for reference, people get head scans with braces pretty regularly and it's not considered a safety issue. Braces and rings can affect image quality though so that's usually the concern. So if the ring is near the body part being imaged you'd probably be asked to remove it because they'll easily cause undesired issues (in, say, roughly a 3-6inch radius) that can result in images that radiologists deem unusable for making a diagnosis.