We present an analysis of , O iii88, N ii122, and far-infrared (FIR) fine-structure line observations obtained with Herschel/PACS, for ∼240 local luminous infrared galaxies (LIRGs) in the Great Observatories All-sky LIRG Survey. We find pronounced declines ("deficits") of line-to-FIR continuum emission for N ii122, , and as a function of FIR color and infrared luminosity surface density, . The median electron density of the ionized gas in LIRGs, based on the N ii122/N ii205 ratio, is = 41 cm−3. We find that the dispersion in the deficit of LIRGs is attributed to a varying fractional contribution of photodissociation regions (PDRs) to the observed emission, f( ) = / , which increases from ∼60% to ∼95% in the warmest LIRGs. The / ratio is tightly correlated with the PDR gas kinetic temperature in sources where is not optically thick or self-absorbed. For each galaxy, we derive the average PDR hydrogen density, , and intensity of the interstellar radiation field, G, in units of and find G/ ratios of ∼0.1-50 cm3, with ULIRGs populating the upper end of the distribution. There is a relation between G/ and , showing a critical break at 5 × 1010 L kpc−2. Below , G/ remains constant, 0.32 cm3, and variations in are driven by the number density of star-forming regions within a galaxy, with no change in their PDR properties. Above , G/ increases rapidly with , signaling a departure from the typical PDR conditions found in normal star-forming galaxies toward more intense/harder radiation fields and compact geometries typical of starbursting sources.