Protein-RNA interactions are integral to the biological functions of RNA. It is well recognized that molecular dynamics (MD) simulations of protein-RNA complexes are more challenging than those of each component. The difficulty arises from the strong electrostatic interactions and the delicate balance between various types of physical forces at the interface. Previously, MD simulations of protein-RNA complexes have predominantly employed fixed-charge force fields. Although force field modifications have been developed to address problems identified in the simulations, some protein-RNA structures are still hard to reproduce by simulations. Here, we present MD simulations of two representative protein-RNA complexes using the AMOEBA polarizable force field. The van der Waals parameters were refined to reproduce accurate quantum-mechanical data of base-base and base-amino acid interactions. It was found that the refined parameters produced a more stable hydrogen-bond network in the interface. One of the complexes remained stable during the short simulations, whereas it could quickly break down in previous simulations using fixed-charge force fields. There was reversible breaking and formation of hydrogen bonds that are observed in the crystal structure, which may indicate the difference in solution and crystal structures. While further improvement and validation of the force fields are still needed, this work demonstrates that polarizable force fields are promising for the study of protein-RNA complexes.