I'm going to assume that the rings are part of his costume design/are essential to his character, since the obvious solution otherwise would be to just have him not wear them on stage.
In regards to the taping, are you just trying to tape them to his fingers? If so, I would suggest actually wrapping some tape and or moleskin around the bottom of the ring itself to serve as a size adjustor. If it fits more snugly on his finger it should lessen the chance of it sliding off and going flying.