Opioid addiction is a complex phenomenon with genetic, social, and other components. Due to such complexity, it is difficult to interpret the outcome of clinical studies, and thus, mutations found in individuals with these addictions are still not indisputably classified as opioid addiction-causing variants. Here, we computationally investigated two such mutations, A6V and N40D, found in the mu opioid receptor gene OPRM1. The mutations are located in the extracellular domain of the corresponding protein, which is important to the hetero-dimerization of OPRM1 with the delta opioid receptor protein (OPRD1). The hetero-dimerization of OPRD1–OPRM1 affects the signaling pathways activated by opioids and natural peptides and, thus, could be considered a factor contributing to addiction. In this study, we built four 3D structures of molecular pathways, including the G-protein signaling pathway and the β-arrestin signaling pathway of the heterodimer of OPRD1–OPRM1. We also analyzed the effect of mutations of A6V and N40D on the stability of individual OPRM1/OPRD1 molecules and the OPRD1–OPRM1 heterodimer with the goal of inferring their plausible linkage with opioid addiction. It was found that both mutations slightly destabilize OPRM1/OPRD1 monomers and weaken their association. Since hetero-dimerization is a key step for signaling processes, it is anticipated that both mutations may be causing increased addiction risk.