The standard LU factorization-based solution process for linear systems can be enhanced in speed or accuracy by employing mixed precision iterative refinement. Most recent work has focused on dense systems. We investigate the potential of mixed precision iterative refinement to enhance methods for sparse systems based on approximate sparse factorizations. In doing so we first develop a new error analysis for LU-and GMRES-based iterative refinement under a general model of LU factorization that accounts for the approximation methods typically used by modern sparse solvers, such as low-rank approximations or relaxed pivoting strategies. We then provide a detailed performance analysis of both the execution time and memory consumption of different algorithms, based on a selected set of iterative refinement variants and approximate sparse factorizations. Our performance study uses the multifrontal solver MUMPS, which can exploit block low-rank (BLR) factorization and static pivoting. We evaluate the performance of the algorithms on large, sparse problems coming from a variety of real-life and industrial applications showing that the proposed approach can lead to considerable reductions of both the time and memory consumption.