Radio waves are non-ionizing, which means they cannot cause what is called "radiation poisoning".
Right... gamma rays are photons. The electromagnetic waves that the router emits are photons. Visible light is also composed of photons. The two core differences are the amount of energy each individual photon carries and the density of photons. In incredibly high densities, low energy photons can ionize... but really very few things besides specialized lasers can generate that kind of density. Alternatively, once you get into higher photon energies, like gamma rays and perhaps the higher end of x-rays, a single photon is powerful enough to ionize -- knock an atom in a solid into a higher energy state / knock it out of its place in the solid structure. This is primarily how radiation damages solid materials, including biological tissue.
Below the ionization threshold, all the EM radiation can do is heat up materials slightly and possibly in some special circumstances disrupt other electrical processes.