Not sure about the structural integrity of the towel rings. What's the diameter of the wood, not the ring but the loop part of the ring. Eg how thick is the wood? Also is it oak carved from a single piece of wood, layered, etc? Keep in mind the ring was designed to hold a towel, not a person. You might be just fine, but definately do some strength tests with them before trusting them to hold your weight. I can pretty much guarantee the manufacturer will tell you that they won't hold that much weight from a liability standpoint.
The grip tape is used only if the rings are used solely for conditioning purposes. If the rings are to be swung on, then you don't want grip tape cause you'll just mangle the grip tape, but then I wouldn't want to swing on towel rings anyway.