The Forward Physics Facility (FPF), planned to operate near the ATLAS interaction point at the LHC, offers exciting new terrain to explore neutrino properties at TeV energy scales. It will reach an unprecedented regime for terrestrial neutrino experiments and provide the opportunity to reveal new physics of neutrinos at higher energy scales. We demonstrate that future detectors at the FPF have the potential to discover new mediators that couple predominantly to neutrinos, with masses between 0.3 and 20 GeV and small couplings not yet probed by existing searches. Such a neutrinophilic mediator is well motivated for addressing the origin of several neutrino-portal dark matter candidates, including thermal freeze-out and sterile-neutrino dark matter scenarios. Experimentally, the corresponding signatures include neutrino charged-current scattering events associated with large missing transverse momentum, and excessive apparent tau-neutrino events. We discuss the FPF detector capabilities needed for this search, most importantly the hadronic energy resolution.