Grain boundaries (GBs) represent one of the most important types of defects in solids and their instability leads to catastrophic failures in materials. Grain boundaries are challenging for theoretical studies because of their distorted atomic structure. Fortunately, quantum-mechanical methods can reliably compute their properties. We calculate and analyze (tensorial) anisotropic elastic properties of periodic approximants of interface states associated with GBs in one of the most important intermetallic compounds for industrial applications, Ni3Al, appearing in Ni-based superalloys. Focusing on the Σ5(210) GBs as a case study, we assess the mechanical stability of the corresponding interface states by checking rigorous elasticity-based Born stability criteria. The critical elastic constant is found three-/five-fold softer contributing thus to the reduction of the mechanical stability of Ni3Al polycrystals (experiments show their GB-related failure). The tensorial elasto-chemical complexity of interface states associated with the studied GBs exemplifies itself in high sensitivity of elastic constants to the GB composition. As another example we study the impact caused by Si atoms segregating into the atomic layers close to the GB and substituting Al atoms. If wisely exploited, our study paves the way towards solute-controlled design of GB-related interface states with controlled stability and/or tensorial properties.