Flotation tailings (FT), as a product of the exploitation and processing of copper ore, represent a significant environmental and health risk due to the high content of heavy metals and sulfide compounds. Contemporary concepts of sustainable development and circular economy increasingly emphasize the need for rational use of resources and minimization of all types of waste, including mining waste. In this context, the reuse of flotation tailings in the construction industry represents a significant step towards closing the material flow in the mining and construction sectors. In order to reduce the negative impact of FT on the environment, the possibility of its application as a substitute for a portion of cement in the production of concrete was investigated. The main challenge is to reduce the negative impact of sulfides, originating from sulfide compounds, in order to achieve the desired concrete quality. Limestone aggregates of different size fractions (0/4, 4/8, 8/16 mm) were used for sulfide neutralization. Pyrite concentrate was used as a sulfide source, which together with FT provides the mixtures FT-7, FT-14, FT-25, and FT-40, with sulfur contents of 7.56, 13.84, 25.02, and 39.82%, respectively. FT mixtures were used as a substitute for Portland cement (PC) in the preparation of concrete. Test methods included XRD (X-ray diffraction), XRF (X-ray fluorescence), SEM (scanning electron microscopy), LP (leaching procedure), TCLP (toxicity characterization leaching procedure), assessment of acid eluate generation potential (AP—acid potential, NP—neutralization potential, and NNP—net neutralization potential), NEN (determination of heavy metals in cured concrete eluate), and UCS (uniaxial compressive strength of cured concrete). The results showed that the chemical characteristics of FT, as well as the chemical and mechanical properties of hardened concrete, allow the efficient use of these tailings in concrete mixes, which significantly utilizes FT, reduces the generation of mining waste, and contributes to the reduction of the negative impact on the environment and achieving sustainable development in mining.