MRI isn't the same thing as an inductive charger. The field is much stronger in an MRI scanner, and the main field is static. There are two dangers: having metal objects pulled by the strong static main field, and heating from the varying gradient fields during scanning. Non-ferrous metal generally isn't much of a problem, but if it's easy to remove you might as well. When I started doing MRI research we emptied our pockets but went into the scanner in our regular clothes, jean rivets, zippers, whatever. You'd suggest women might want to take off underwire bras, but it wasn't insisted upon. Regulation creep now means most centres insist on subjects being stripped down to scrubs. Most piercings, dental work, etc. are okay, unless they're near the area being imaged (they can distort the images).
Inductive chargers are lower field, oscillating, and tuned to match a specific receiver coil geometry. You're unlikely to get any significant power transfer to something random like a wedding ring. Especially since the receiver coil is likely much smaller than a finger; you wouldn't want to carry something that big attached to your cell phone.