In this paper, the pinning and depinning mechanism of the contact line during droplet evaporation on chemically stripe-patterned surfaces is numerically investigated using a thermal multiphase lattice Boltzmann (LB) model with liquid-vapor phase change. A local force balance in the context of diffuse interfaces is introduced to explain the equilibrium states of droplets on chemically patterned surfaces. It is shown that when the contact line is pinned on a hydrophobic-hydrophilic boundary, different contact angles can be interpreted as the variation of the length of the contact line occupied by each component. The stick-slip-jump behavior of evaporating droplets on chemically patterned surfaces is well captured by the LB simulations. Particularly, a slow movement of the contact line is clearly observed during the stick (pinning) mode, which shows that the pinning of the contact line during droplet evaporation on chemically stripe-patterned surfaces is actually a dynamic pinning process and the dynamic equilibrium is achieved by the self-adjustment of the contact lines occupied by each component. Moreover, it is shown that when the surface tension varies with the temperature, the Marangoni effect has an important influence on the depinning of the contact line, which occurs when the horizontal component (toward the center of the droplet) of the force caused by the Marangoni stress overcomes the unbalanced Young's force toward the outside.