Umm, pray tell, what has heating up (using microwaves) got to do with ionization? Radiation sickness happens because key elements of cells get damaged. The chemistry of our cells is not designed to deal with intense interference with structures of various complex molecules. Ionizing radiation is called such because it can dissociate molecules into ions - that's a rather significant alteration, the original molecule ceases to exist.
The sunburn is a very specific reaction to DNA damage and only DNA damage. The DNA is sensitive to non-ionizing UV-B radiation; new chemical bonds are formed when a photon is absorbed and doesn't get converted to heat. This happens without ionization having occurred. The DNA is simply, by design, susceptible to this kind of radiation, and the direct damage mechanism of DNA is a part of a large family of photochemical reactions. In terms of sensitivity per photon, the DNA is about 2 orders of magnitude less sensitive (~0.1% eff.) than the silver halide you'd find in a film emulsion (~20% eff. undoped).