We report on new sensitive CO J = 6-5 line observations of several luminous infrared galaxies (LIRGs; L {sub IR}(8-1000 {mu}m) {approx}> 10{sup 11} L {sub sun}), 36% (8/22) of them ultraluminous infrared galaxies (ULIRGs) (L {sub IR}>10{sup 12} L {sub sun}), and two powerful local active galactic nuclei (AGNs)-the optically luminous QSO PG 1119+120 and the powerful radio galaxy 3C 293-using the James Clerk Maxwell Telescope on Mauna Kea in Hawaii. We combine these observations with existing low-J CO data and dust emission spectral energy distributions in the far-infrared-submillimeter from the literature to constrain the properties of the star-forming interstellar medium (ISM) in these systems. We then build the first local CO spectral line energy distributions (SLEDs) for the global molecular gas reservoirs that reach up to high J-levels. These CO SLEDs are neither biased by strong lensing (which affects many of those constructed for high-redshift galaxies), nor suffer from undersampling of CO-bright regions (as most current high-J CO observations of nearby extended systems do). We find: (1) a significant influence of dust optical depths on the high-J CO lines, suppressing the J = 6-5 line emission in some of the most IR-luminous LIRGs, (2) low global CO line excitation possible even in vigorously star-forming systems, (3) the first case of a shock-powered high-excitation CO SLED in the radio galaxy 3C 293 where a powerful jet-ISM interaction occurs, and (4) unusually highly excitated gas in the optically powerful QSO PG 1119+120. In Arp 220 and possibly other (U)LIRGs very faint CO J = 6-5 lines can be attributed to significant dust optical depths at short submillimeter wavelengths immersing those lines in a strong dust continuum, and also causing the C{sup +} line luminosity deficit often observed in such extreme starbursts. Re-analysis of the CO line ratios available for submillimeter galaxies suggests that similar dust opacities also may be present in these high-redshift starbursts, with genuinely low excitation of large amounts of SF-quiescent gas being the only other possibility for their often low CO (high-J)/(low-J) line ratios. We then present a statistical method of separating these two almost degenerate possibilities, and show that high dust optical depths at submillimeter wavelengths can impede the diagnostic potential of submillimeter/IR lines (e.g., starbursts versus AGNs as gas excitation agents), which is of particular importance for the upcoming observations of the Herschel Space Observatory and the era of ALMA.