Geopolymers are environmentally friendly materials made from industrial solid waste with high silicon and aluminum contents, and municipal solid waste incineration fly ash (MFA) contains active ingredients such as Si, Al and Ca. According to this fact, a green and low-carbon geopolymer concrete was prepared using MFA as a partial replacement for metakaolin in this study. The mechanical properties of the MFA geopolymer concrete (MFA-GPC) were investigated through a series of experiments, including a compressive strength test, splitting tensile strength test, elastic modulus test and three-point bending fracture test. The effect of the MFA replacement ratio on the microstructure of MFA-GPC was investigated by SEM test, XRD analysis and FTIR analysis. MFA replacement ratios incorporated in GPC were 5%, 10%, 15%, 20%, 25%, 30%, 35% and 40% by replacing metakaolin with equal quality in this study. In addition, toxic leaching tests of MFA and MFA-GPC were performed by ICP-AES to evaluate the safety of MFA-GPC. The results indicated that the mechanical properties of MFA-GPC decreased with the increase of the MFA replacement ratio. Compared with the reference group of GPC without MFA, the maximum reduction rates of the cubic compressive strength, splitting tensile strength, axial compressive strength, elastic modulus, initiation fracture toughness, unstable fracture toughness and fracture energy of MFA-GPC were 83%, 81%, 78%, 93%, 77%, 73% and 61%, respectively. The microstructure of MFA-GPC was porous and carbonized; however, the type of hydrated gel products was still a calcium silicoaluminate-based silicoaluminate gel. Moreover, the leaching content of heavy metals from MFA-GPC was lower than that of the standard limit. In general, the appropriate amount of MFA can be used to prepare GPC, and its mechanical properties can meet the engineering requirements, but the amount of MFA should not be too high.