The paper presents the numerical and experimental results of research aimed at determining the influence of hardness in the range of 50–90 Shore A of layered tools composed of elastomeric materials on the possibility of forming Inconel 625 nickel-based alloy sheets. A stamping die composed of 90MnCrV8 steel (hardness 60HRC) was designed for forming embosses in drawpieces, ensuring various stress states on the cross-section of the formed element. The principle of operating the stamping die was based on the Guerin method. The finite-element-based numerical modelling of the forming process for various configurations of polyurethane inserts was also carried out. The drawpieces obtained through sheet forming were subjected to geometry tests using optical 3D scanning. The results confirmed that, in the case of forming difficult-to-deform Inconel 625 Ni-based alloy sheets, the hardness of the polyurethane inserts significantly affected the geometric quality of the obtained drawpieces. The assumptions determined in numerical simulations were verified in experimental studies. Based on the test results, it was concluded that the selection of polyurethane hardness should be determined by the shape of the formed element. Significant nonuniform sheet metal deformations were also found, which may pose a problem in the process of designing forming tools and the technology of the plastic forming of Inconel 625 Ni-based alloy sheets.