We consider infinite-dimensional Bayesian linear inverse problems constrained by PDEs where data are collected via moving sensors. In this context, we develop a framework for finding sensor paths that optimize the posterior uncertainty in a prediction functional of the inversion parameters. The mathematical formulations are first established in an infinite-dimensional setting. Subsequently, we consider discretization of the problem and present a scalable computational framework for finding optimal sensor paths. We also present illustrative numerical results, within the context of a model inverse problems governed by an advection diffusion equation.