The evolutionary mechanism underlying Type Ia supernova explosions remains unknown. Recent efforts to constrain progenitor models based on the influence that their high energy emission would have on the interstellar medium (ISM) of galaxies have proven successful. For individual remnants, Balmer-dominated shocks reveal the ionization state of hydrogen in the immediately surrounding gas. Here we report deep upper limits on the temperature and luminosity of the progenitors of four Type Ia remnants with associated Balmer filaments: SN 1006, 0509-67.5, 0519-69.0, and DEM L71. For SN 1006, existing observations of helium line emission in the diffuse emission ahead of the shock provide an additional constraint on the helium ionization state in the vicinity of the remnant. Using the photoionization code Cloudy, we show that these constraints exclude any hot, luminous progenitor for SN 1006, including stably hydrogen or helium nuclear-burning white dwarfs, as well as any Chandrasekhar-mass white dwarf accreting matter at $gtrsim 9.5times10^{-8}M_{odot}/$yr via a disk. For 0509-67.5, the Balmer emission alone rules out any such white dwarf accreting $gtrsim 1.4times10^{-8}M_{odot}/$yr. For 0519-69.0 and DEM L71, the inferred ambient ionization state of hydrogen is only weakly in tension with a recently hot, luminous progenitor, and cannot be distinguished from e.g., a relatively higher local Lyman continuum background, without additional line measurements. Future deep spectroscopic observations will resolve this ambiguity, and can either detect the influence of any luminous progenitor or rule out the same for all resolved SN Ia remnants.
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