Waste tires cause serious environmental problems, such as pollution, which can be reduced by mechanically grinding rubber tires into powder to obtain rubber particles. These components are then used in equal mass (20%) to replace fine aggregate to manufacture rubber concrete (RC), a new type of environmentally friendly building material. This study explored the performance of RC and different volume fractions (0.5%, 1.0% and 1.5%) basalt-polypropylene fiber-reinforced rubber concrete (BPRC) under elevated temperatures through a series of experiments on mass, relative dynamic elastic modulus, static compressive strength, and dynamic compressive strength. The experimental results showed that with increasing temperature, mass loss in RC and BPRC gradually increased, their relative dynamic elastic moduli slowly decreased. Their static compressive and dynamic compressive strengths reflected different degrees of loss, the strength loss temperature node of pure rubber concrete is earlier and the loss degree is more. The best performance was shown by the BPRC with 1% and 1.5% volume ratios of basalt and polypropylene fibers, respectively. Mixing basalt and polypropylene fibers into RC is beneficial to improvements in its performance after an increase in temperature.