We establish quasi-two-dimensional thin films of iron-based superconductors (FeSCs) as a new hightemperature platform for hosting intrinsic time-reversal-invariant helical topological superconductivity (TSC). Based on the combination of Dirac surface state and bulk extended s-wave pairing, our theory should be directly applicable to a large class of experimentally established FeSCs, opening a new TSC paradigm. In particular, an applied electric field serves as a "topological switch" for helical Majorana edge modes in FeSC thin films, allowing for an experimentally feasible design of gate-controlled helical Majorana circuits. Applying an in-plane magnetic field drives the helical TSC phase into a higher-order TSC carrying corner-localized Majorana zero modes. Our proposal should enable the experimental realization of helical Majorana fermions.