<p>This work focuses on the nonlinear dynamic response of composite beams reinforced with carbon nanotubes (CNTs), and subjected to a moving mass, accounting for different reinforcement distributions. The Hamilton’s principle is here combined to the Euler–Bernoulli beam theory, including a von Kármán nonlinearity in the kinematic assumptions, to determine the nonlinear governing equation of the problem. Unlike previous finite element or semi-analytical studies that primarily relied on numerical discretization or linearized approximations, the present work provides a fully analytical treatment of the nonlinear dynamics by employing the Galerkin decomposition and method of multiple time scales (MMS) to investigate superharmonic and subharmonic resonance coupled with internal resonance. This allows a direct insight into the nonlinear resonance behavior, frequency–amplitude dependence, and parametric influence of moving-mass velocity, magnitude, and position on CNT-reinforced composite (CNTRC) beams—effects that have not been explicitly characterized in earlier finite element method (FEM)-based studies. The results yield useful benchmarks for future computational validation and material–structural design optimization.</p>