The coupling of shear and fast Alfvén waves in the vicinity of a magnetic X-point is studied for the case in which a weak longitudinal guide field $B_{z0}$ is present, with variations in the longitudinal direction and plasma pressure effects neglected. It is shown that an analytical solution for the shear wave in the limit of ideal magnetohydrodynamics (MHD) and $B_{z0}\,{=}\,0$ exhibits phase mixing and equilibration of the field and kinetic energy at a rate that increases with the spatial extent of the initial disturbance. Equations describing perturbatively the pumping of fast waves by shear waves in the presence of finite $B_{z0}$ are derived and solved numerically, taking into account resistivity and electron inertial effects. It is shown that shear wave energy is most rapidly and efficiently converted to plasma kinetic energy when the collisionless skin depth exceeds the resistive scale length. The conversion of incompressible MHD modes to compressible modes at X-points provides a possible mechanism for solar coronal heating, and mode conversion processes of this type are also likely to occur in tokamak plasmas with X-points.