In recent times, the applications of fiber-reinforced recycled aggregate concrete (FRAC) in practical engineering have gained greater popularity due to its superior mechanical strength and fracture properties. To apply FRAC in buildings and other infrastructures, a thorough understanding of its residual mechanical properties and durability after exposure to fire is highly important. According to the established research, the properties and volume fractions of reinforcing fiber materials, replacement levels of recycled concrete aggregate (RCA), and heating condition would affect the thermal–mechanical properties of FRAC. This review paper aims to present a thorough and updated review of the mechanical performance at an elevated temperature and post-fire durability of FRAC reinforced with various types of fiber material, specifically steel fiber (SF), polypropylene (PP) fiber, and basalt fiber (BF). More explicitly, in this review article the residual mechanical properties of FRAC, such as compressive strength, splitting tensile capacity, modulus of elasticity, mass loss, spalling, and durability after exposure to elevated temperatures, are discussed. Furthermore, this study also encompasses the relationship among the dosages of fibers, replacement levels of recycled aggregate, and the relative residual mechanical properties of FRAC that would help in the optimum selection of the fiber content. Conclusively, this study elaborately reviews and summarizes the relevant and recent literature on recycled aggregate concrete containing SF, PP fiber, and BF. The study further provides a realistic comparison of these fibers in terms of the residual mechanical performance and durability of FRAC that would help in their future enhancements and applications in practical engineering.