In the framework of the third-order shear deformation theory, the coupled size-dependent nonlinear dynamics of a viscoelastic extensible microbeam is investigated on the basis of the modified couple stress theory. Taking into account both the transverse and axial displacements as well as inertia together with the rotation and rotary inertia, the viscous parts of the classical and nonclassical stress tensors are calculated and the corresponding work terms are obtained. These viscous work terms are balanced by the kinetic energy of the system, the size-dependent potential energy, and the work of an external excitation force. This operation leads to coupled expressions for the transverse, rotational, and longitudinal motions. Retaining the rotation, rotary inertia as well as translational displacements and inertia (for the transverse and longitudinal motions), three high-dimensional reduced-order models are obtained for the transverse, longitudinal, and rotational motions employing a weighted-residual method. These three size-dependent viscoelastically coupled models are solved via use of the pseudo-arclength continuation method in conjunction with the backward differentiation formula (BDF). It is shown that the viscoelastic model is an amplitude-dependent energy dissipation mechanism which gives more accurate results compared to an elastic model. More specifically, as the forcing amplitude is increased, the response of the elastic system deviates from that of the viscoelastic system.