To improve the long-term performance of concrete engineering in high-altitude areas, waste tire rubber was added to a concrete mix, and freeze–thaw and impact tests were conducted. The effects of waste tire rubber with different particle sizes (10, 20, 30 mesh) and freeze–thaw cycles (0, 25, 50, 75, 100, 125) on the dynamic mechanical properties of concrete materials were studied. The stress–strain curves, peak stress, and fracture morphology of the specimens were analyzed. The microstructure changes of the specimens were also analyzed using scanning electron microscopy (SEM). The results showed the following: (1) Both macroscopic and microscopic analysis results showed that the internal damage of rubber concrete specimens was smaller after freeze–thawing, and the integrity was better after impact, maintaining a loose but not scattered state. The addition of waste tire rubber significantly improved the material’s impact resistance to a certain extent. (2) As the impact pressure increased, the strain rate of the specimens increased linearly, and the dynamic peak stress was linearly positively correlated with the strain rate. (3) After 125 freeze–thaw cycles, the peak stress of the specimens with 30-mesh added rubber decreased significantly less than that of ordinary concrete under 0.3, 0.45, and 0.6 MPa impact pressure. The dynamic peak stress was higher than that of specimens with 10-mesh and 20-mesh added rubber, and the addition of 30-mesh rubber significantly improved the frost resistance and impact resistance of concrete materials. This study can provide new ideas for the engineering application of rubber concrete.