In this paper, we extend our previous immersed boundary (IB) projection method for 2D inextensible vesicle in unsteady Stokes flow (Ong and Lai, 2020 ) to 3D incompressible interface as a prototype of vesicles. In spite of similar numerical algorithm to the 2D case, the present 3D numerical implementation is far from straightforward. An incompressible interface immersed in Newtonian fluid must ensure both conservative constraints associated with local fluid volume and local interfacial area. To accomplish this, we triangulate the incompressible interface, whereby the unknown elastic tension is defined on the triangles, while the associated tension force term is defined at vertices (Lagrangian markers). Consistent approximations of surface differential operators on a triangulated interface are developed accordingly. An extra scalar field is introduced in the IB projection approach so that the fluid pressure and interfacial elastic tension are efficiently solved in a divide-and-conquer manner. As a result, the local interfacial and volumetric incompressibilities can be ideally satisfied simultaneously. A series of numerical tests on the present scheme is performed to verify the robustness and applicability of the method. We first carry out the convergence study of the solution variables defined on the fluid and interface. We then study the shear-induced deformations of incompressible interface for various initial configurations. At last, the bending force and the penalty force for improved volume conservation are further incorporated into our method to simulate an oblate vesicle dynamics in quiescent flow.