We investigate the dynamics of a monodisperse polymer melt that is confined to the vicinity of a plane by a harmonic potential. Polymer molecules are represented by harmonic bead-spring chains with fluctuating bead mobility tensors. Mobility fluctuation rates are determined self-consistently from the chain dynamics. The calculation of the viscoelastic shear modulus is mapped onto the solution of a dynamically disordered random walk, in which a walker hops among lattice sites with fluctuating hopping rates. The relevant random walk problems are solved within the effective medium approximation, yielding the shear modulus and coefficient of shear viscosity as functions of chain length, entanglement molecular weight, and strength of the confining potential. Calculations illustrate the competition between entanglement effects, which retard chain dynamics, and the effect of the external field, which promotes relaxation.