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We present an analysis of [OI]63, [OIII]88, [NII]122 and [CII]158 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 (GOALS). We find pronounced declines -deficits- of line-to-FIR-continuum emission for [NII]122, [OI]63 and [CII]158 as a function of FIR color and infrared luminosity surface density, $\Sigma_{\rm IR}$. The median electron density of the ionized gas in LIRGs, based on the [NII]122/[NII]205 ratio, is $n_{\rm e}$ = 41 cm$^{-3}$. We find that the dispersion in the [CII]158 deficit of LIRGs is attributed to a varying fractional contribution of photo-dissociation-regions (PDRs) to the observed [CII]158 emission, f([CII]PDR) = [CII]PDR/[CII], which increases from ~60% to ~95% in the warmest LIRGs. The [OI]63/[CII]158PDR ratio is tightly correlated with the PDR gas kinetic temperature in sources where [OI]63 is not optically-thick or self-absorbed. For each galaxy, we derive the average PDR hydrogen density, $n_{\rm H}$, and intensity of the interstellar radiation field, in units of G$_0$, and find G$_0$/$n_{\rm H}$ ratios ~0.1-50 cm$^3$, with ULIRGs populating the upper end of the distribution. There is a relation between G$_0$/$n_{\rm H}$ and $\Sigma_{\rm IR}$, showing a critical break at $\Sigma_{\rm IR}^{\star}$ ~ 5 x 10$^{10}$ Lsun/kpc$^2$. Below $\Sigma_{\rm IR}^{\star}$, G$_0$/$n_{\rm H}$ remains constant, ~0.32 cm$^3$, and variations in $\Sigma_{\rm IR}$ are driven by the number density of star-forming regions within a galaxy, with no change in their PDR properties. Above $\Sigma_{\rm IR}^{\star}$, G$_0$/$n_{\rm H}$ increases rapidly with $\Sigma_{\rm IR}$, signaling a departure from the typical PDR conditions found in normal star-forming galaxies towards more intense/harder radiation fields and compact geometries typical of starbursting sources., Comment: 23 pages, 12 figures. Submitted to ApJ, including 2nd round of referee comments. Data tables can be found at: http://goals.ipac.caltech.edu/ and will also appear on the electronic version of the journal