Neutral Fe lines in metal-poor stars yield conflicting abundances depending on whether and how deviations from local thermodynamic equilibrium (LTE) are considered. We have collected new high-resolution and high signal-to-noise ultraviolet (UV) spectra of three warm dwarf stars with [Fe/H]~-2.9 with the Space Telescope Imaging Spectrograph on the Hubble Space Telescope. We locate archival UV spectra for three other warm dwarfs with [Fe/H]~-3.3, -2.2, and -1.6, supplemented with optical spectra for all six stars. We calculate stellar parameters using methods that are largely independent of the spectra, adopting broadband photometry, color-temperature relations, Gaia parallaxes, and assumed masses. We use the LTE line analysis code MOOG to derive Fe abundances from hundreds of FeI and FeII lines with wavelengths from 2290 to 6430{AA}. The [Fe/H] ratios derived separately from FeI and FeII lines agree in all six stars, with [FeII/H]-[FeI/H] ranging from +0.00+/-0.07 to -0.12+/-0.09dex, when strong lines and FeI lines with lower excitation potential <1.2eV are excluded. This constrains the extent of any deviations from LTE that may occur within this parameter range. While our result confirms non-LTE calculations for some warm, metal-poor dwarfs, it may not be generalizable to more metal-poor dwarfs, where deviations from LTE are predicted to be larger. We also investigate trends of systematically lower abundances derived from FeI lines in the Balmer continuum region (~3100-3700{AA}), and we conclude that no proposed explanation for this effect can fully account for the observations presently available.