Ultrafast charge transfer in van der Waals (vdW) heterostructures enables efficient control of two-dimensional material properties through strong optical absorption and subsequent carrier transfer. Here, using real-time time-dependent density functional theory coupled to molecular dynamics, we investigated the nonequilibrium dynamics of charge-density-wave (CDW) melting in 1T-TaS2 triggered by ultrafast charge transfer in 1T-TaS2/MoSe2 or WSe2 heterostructures. Despite the fast and sufficient charge transfer from the MoSe2 (or WSe2) "electrode" to the 1T-TaS2 layer, the electronic excitation of the vdW heterostructure does not lead to the nonthermal CDW transition of 1T-TaS2. Instead, the TaS2 lattice is heated by carrier-lattice scattering, leading to thermal CDW melting at high ionic temperatures. The lack of nonthermal melting follows from the fact that the time scale of carrier recombination in 1T-TaS2 is similar to or faster than that of charge transfer. These findings provide physical insights into understanding the CDW melting dynamics in vdW heterostructures under nonequilibrium conditions.