The future circular electron-positron collider will provide collisions at four interaction points (IPs) along a 91-km ring, with beam energies ranging from 45.6 GeV (Z pole) to 182.5 GeV ( threshold). Several processes, induced by both the collisions at the IPs and the circulating beams, contribute to the radiation environment of the experimental insertion regions, exposing the detector, machine equipment, and electronics. Quantifying these radiation fields and their effects is essential to guide design strategies. In this work, the Monte Carlo code fluka is employed to simulate the propagation of radiation showers from both IP-driven and beam-driven sources, including radiative Bhabha scattering, incoherent pair creation, beamstrahlung, synchrotron radiation, and beam-gas interactions. Power deposition, equivalent fluences, and total ionizing dose are estimated to assess the exposure of sensitive components and to characterize tunnel radiation levels at the Z-pole and -threshold modes. The results show that incoherent pairs dominate the detector environment, particularly in the tracker, while radiative Bhabha losses in the superconducting final-focus quadrupoles (FFQs) require shielding to ensure safe magnet operation. Tunnel radiation levels exceed the tolerance of standard equipment at both operational modes, being dominated by radiative Bhabha and collimation losses at the Z pole, and by synchrotron radiation at the threshold. While a tungsten layer is shown to effectively protect the FFQs, the high radiation levels elsewhere indicate the need for additional shielding in dipoles and drifts, as well as dedicated absorbers downstream of the IPs and collimators.