Motivated by recent reports of H_{3}LiIr_{2}O_{6} as a spin-orbital-entangled quantum liquid, we investigate via a combination of density functional theory and nonperturbative exact diagonalization the microscopic nature of its magnetic interactions. We find that while the interlayer O─H─O bond geometry strongly affects the local magnetic couplings, these bonds are likely to remain symmetrical due to large zero-point fluctuations of the H positions. In this case, the estimated magnetic model lies close to the classical tricritical point between ferromagnetic, zigzag, and incommensurate spiral orders, what may contribute to the lack of magnetic ordering. However, we also find that substitution of H by D (deuterium) as well as disorder-induced inhomogeneities destabilizes the O─H or D─O bonds, modifying strongly the local magnetic couplings. These results suggest that the magnetic response in H_{3}LiIr_{2}O_{6} is likely sensitive to both the stoichiometry and the microstructure of the samples and emphasize the importance of a careful treatment of hydrogen for similar systems.