No. Even if you take on board somethinghere's correct answer and instead try to build the waveguide from wholly lossless, but physical, materials, the loss will still be there. The loss in a ring resonator is mostly bend loss, which arises even when the ring is made from perfectly lossless materials.
Bend loss arises because the waveguide is no longer translationally invariant (owing to the bend) so that the bound eigenmodes of the formerly straight waveguide are no longer eigenmodes but instead have a nonzero coupling co-efficient with the radiation field.
So the loss from the ring is radiated away from the ring. This is why a very tight ring bearing visible light glows.
Question from OP
At the risk of asking a turtles all the way down question, why does this coupling begin when the waveguide is curved? More specifically, why does the classical model of total reflection no longer apply?
This is an excellent question that exposes the tacit assumption I made in my answer above, i.e. that I was assuming a ring resonator where the EM field propagates power around the ring's circumference in a vibration often called a "whispering gallery mode". The answer above can be summarized as, "whispering gallery modes are never true modes owing to an unavoidable, everpresent coupling to the radiation field", and the answer still stands. Indeed, I believe this tacit assumption was also made in your answer, because I get the impression that you are talking about modes where energy moves around the ring's circumference.
However, why, as you ask, doesn't the classical notion of total internal reflexion work with bends? Let's anticipate and dispense with a sly answer before getting to the real one: one could say that the the usual model of TIR assumes Snell's law and thus plane waves, so that bends violate these assumptions. OK, but your question then becomes, "Why can't we derive a classical model for TIR for bends?".
The answer to this last one is that we can indeed derive such a model, and the ring structure does have lossless modes corresponding to the new, rotational cylindrical symmetry. If you write down Maxwell's equations in cylindrical co-ordinates, these lossless modes pop out (with Hankel function dependencies of the field polar components on the radial co-ordinate). These modes are akin to the lossless vibrations of a spherical cavity resonator. However, in these modes, there is no energy transfer around the ring's circumference. There is no nett power transfer at all: the device is a purely reactive device where energy shuttles to and fro outwards then back inwards across the ring, with no nett loss each cycle.
So there is a model of lossless TIR for bent waveguides, but the propagation directions are radial, not circumferential, so there is no transfer of energy around the ring as in a ring resonator. This last point is important in most laboratory ring resonators, which are excited by a coupled optical fiber with field propagating tangentially to the ring.