This paper investigates the effects of surface residual stress and surface elasticity modulus on the flutter instability of a carbon nanotube (CNT) conveying magnetic fluid and resting on a Pasternak elastic foundation. The dynamic modeling is performed using Eringen’s nonlocal elasticity theory based on the Euler–Bernoulli beam model, with a fixed-end (cantilever) boundary condition, under loading induced by fluid flow and a magnetic field. First, the potential energy functions associated with fluid–nanotube interaction forces, magnetic field forces, Pasternak foundation forces, as well as the kinetic and strain energies of the nanotube, are formulated. The governing classical equations are then derived using the energy method and Hamilton’s principle. By combining the resulting equations with Eringen’s nonlocal theory and the Gurtin–Murdoch surface elasticity theory, the differential equation governing the transverse vibration of the nanotube is obtained. The Galerkin method, employing four basis functions, is used to obtain an approximate solution. Throughout the simulations, as the fluid flow velocity increases, flutter instability of the nanotube occurs when the real part of the eigenvalue corresponding to the third vibration mode becomes zero, indicating the onset of self-excited vibration. The results demonstrate that increasing the surface elasticity modulus and decreasing the surface residual stress raise the critical flutter velocity and expand the stability range of the nanotube.