Stacking of atomically thin layers of two-dimensional materials has revealed extraordinary physical phenomena owing to van der Waals (vdW) interaction at the interface. However, most of the studies focused on the transition metal dichalcogenide (TMD)/TMD heterostructure, while the interlayer coupling of the TMD/hexagonal boron nitride (h-BN) heterostructure has not been extensively explored despite its importance. In this study, the temperature-dependent interlayer coupling is demonstrated in a heterostructure of molybdenum disulfide (MoS2) and h-BN. The interface between MoS2 and the insulating substrate exerts a significant spectroscopic impact on MoS2 through substrate-induced local strain, charged impurity, and vdW interactions. Under non-resonant conditions, temperature-dependent peak shifts in Raman and photoluminescence (PL) spectra of MoS2 reveal the evolution of interlayer coupling. Phonon frequencies and PL peak energies at different temperatures demonstrate how substrate-induced strain, impurity, and vdW interactions at the interface influence phonon vibration and excitonic transition of MoS2. Under resonant conditions at low temperature, anomalous Raman modes appear in the MoS2/h-BN heterostructure because of the enhanced electron-phonon coupling and vdW interactions. The anomalous Raman modes are quantitatively investigated by the deconvolution of the resonance Raman spectra and described by interlayer coupling at low temperature, in agreement with complementary indications from the temperature-dependent evolution of non-resonant Raman and PL spectra.