We present an abstract framework to study weak convergence of numerical approximations of linear stochastic partial differential equations driven by additive Lévy noise. We first derive a representation formula for the error which we then apply to study space-time discretizations of the stochastic heat equation, a Volterra-type integro-differential equation, and the wave equation as examples. For twice continuously differentiable test functions with bounded second derivative (with an additional condition on the second derivative for the wave equation) the weak rate of convergence is found to be twice the strong rate. The results extend earlier work by two of the authors, as we consider general square-integrable infinite-dimensional Lévy processes and do not require boundedness of the test functions and their first derivative. Furthermore, the present framework is applicable to both hyperbolic and parabolic equations, and even to stochastic Volterra integro-differential equations.