In this paper, we conduct a comprehensive investigation on the thermal transport in one-dimensional (1D) van der Waals (vdW) heterostructures by using non-equilibrium molecular dynamics simulations. It is found that the boron nitride nanotube (BNNT) coating can increase the thermal conductance of inner carbon nanotube (CNT) base by 36%, while the molybdenum disulfide nanotube (MSNT) coating can reduce the thermal conductance by 47%. The different effects of BNNT and MSNT coatings on the thermal transport behaviors of 1D vdW heterostructures are explained by the competition mechanism between improved heat flux and increased temperature gradient in 1D vdW heterostructures. By taking [email protected]@MSNT as an example, thermal transport in 1D vdW heterostructures containing three layers is also investigated. It is found that the coaxial BNNT-MSNT coating can significantly reduce the thermal conductance of inner CNT base by 61%, which is even larger than that of an individual MSNT coating. This unexpected reduction in thermal conductance of [email protected]@MSNT can be explained by the suppression of heat flux arising from the possible compression effect, since BNNT-MSNT coating in [email protected]@MSNT can more significantly suppress the vibration of inner CNT when compared to the individual MSNT coating in [email protected] In addition to the in-plane thermal transport, the interfacial thermal conductance between inner and outer nanotubes in 1D vdW heterostructures is also examined to provide a quantitative understanding of the thermal transport behaviors of1D vdW heterostructures. This work is expected to provide molecular insights into tailoring the heat transport in carbon base 1D vdW heterostructures and thus facilitate their broader applications as thermal interface materials.