We present new data obtained with the Submillimeter Array for a sample of 14 nearby luminous and ultraluminous infrared galaxies. The galaxies were selected to have distances D-L < 200 Mpc and far-infrared luminosities log L-FIR > 11.4. The galaxies were observed with spatial resolutions of order 1 kpc in the COJ = 3-2, COJ = 2-1, (13)COJ = 2-1, and HCO+ J = 4-3 lines as well as the continuum at 880 mu m and 1.3 mm. We have combined our CO and continuum data to measure an average gas-to-dust mass ratio of 120 +/- 28 (rms deviation 109) in the central regions of these galaxies, very similar to the value of 150 determined for the Milky Way. This similarity is interesting given the more intense heating from the starburst and possibly accretion activity in the luminous infrared galaxies compared to the Milky Way. We find that the peak H-2 surface density correlates with the far-infrared luminosity, which suggests that galaxies with higher gas surface densities inside the central kiloparsec have a higher star formation rate. The lack of a significant correlation between total H-2 mass and far-infrared luminosity in our sample suggests that the increased star formation rate is due to the increased availability of molecular gas as fuel for star formation in the central regions. In contrast to previous analyses by other authors, we do not find a significant correlation between central gas surface density and the star formation efficiency, as traced by the ratio of far-infrared luminosity to nuclear gas mass. Our data show that it is the star formation rate, not the star formation efficiency, that increases with increasing central gas surface density in these galaxies.