Using a mathematical model, we investigate the role of hydrodynamic forces on three-dimensional axisymmetric multicomponent vesicles. The equations are derived using an energy variation approach that accounts for different surface phases, the excess energy associated with surface domain boundaries, bending energy and inextensibility. The equations are high-order (fourth order) nonlinear and nonlocal. To solve the equations numerically, we use a nonstiff, pseudo-spectral boundary integral method that relies on an analysis of the equations at small scales. We also derive equations governing the dynamics of inextensible vesicles evolving in the absence of hydrodynamic forces and simulate numerically the evolution of this geometric model. We find that compared with the geometric model, hydrodynamic forces provide additional paths for relaxing inextensible vesicles. The presence of hydrodynamic forces may enable the dynamics to overcome local energy barriers and reach lower energy states than those accessible by geometric motion or energy minimization algorithms. Because of the intimate connection between morphology, surface phase distribution and biological function, these results have important consequences in understanding the role vesicles play in biological processes.