The FLUKA code is used to simulate the residual dose rates around a typical beam absorber considering various scenarios. The latter include carbon, copper, and tungsten as jaw materials, different beam energies, protons, and lead ion beams as well as different irradiation and cooling times. Using the dose rate maximum close to the absorber surface, the study investigates the cooling time dependence for the different scenarios. It is found to be similar for all jaw materials and beam energies. The dose rate scales with energy as E0.83 and with the number of nucleons when comparing proton beam with lead ions. After a sufficiently long cooling time, a few radionuclides produced in the steel tank, such as 56Co, 58Co, 48V, and 54Mn, dominate the dose rate. The study can be easily extended to other materials or irradiation scenarios and can be applied to first evaluations of given accelerator design options.