We report results from a large molecular line survey of luminous infrared galaxies (LIRGs; ) in the local Universe (z ≤ 0.1), conducted during the last decade with the James Clerk Maxwell Telescope and the IRAM 30-m telescope. This work presents the CO and 13CO line data for 36 galaxies, further augmented by multi-J total CO line luminosities available for other infrared (IR) bright galaxies from the literature. This yields a combined sample of N = 70 galaxies with the star formation (SF) powered fraction of their IR luminosities spanning and a wide range of morphologies. Simple comparisons of their available CO spectral line energy distributions (SLEDs) with local ones, as well as radiative transfer models, discern a surprisingly wide range of average interstellar medium (ISM) conditions, with most of the surprises found in the high-excitation regime. These take the form of global CO SLEDs dominated by a very warm (Tkin ≳100 K) and dense (n ≥ 104 cm−3) gas phase, involving galaxy-sized (∼(few) × 109 M⊙) gas mass reservoirs under conditions that are typically found only for ∼(1–3) per cent of mass per typical SF molecular cloud in the Galaxy. Furthermore, some of the highest excitation CO SLEDs are found in ultraluminous infrared galaxies (ULIRGs; LIR ≥ 1012 L⊙) and surpass even those found solely in compact SF-powered hot spots in Galactic molecular clouds. Strong supersonic turbulence and high cosmic ray energy densities rather than far-ultraviolet/optical photons or supernova remnant induced shocks from individual SF sites can globally warm the large amounts of dense gas found in these merger-driven starbursts and easily power their extraordinary CO line excitation. This exciting possibility can now be systematically investigated with Herschel and the Atacama Large Milimeter Array (ALMA). As expected for an IR-selected (and thus SF rate selected) galaxy sample, only few ‘cold’ CO SLEDs are found, and for fewer still a cold low/moderate-density and gravitationally bound state (i.e. Galactic type) emerges as the most likely one. The rest remain compatible with a warm and gravitationally unbound low-density phase often found in ULIRGs. Such degeneracies, prominent when only the low-J SLED segment (J = 1–0, 2–1 and 3–2) is available, advise against using its CO line ratios and the so-called Xco = M(H2)/Lco(1–0) factor as SF mode indicators, a practice that may have led to the misclassification of the ISM environments of IR-selected gas-rich discs in the distant Universe. Finally, we expect that the wide range of ISM conditions found among LIRGs will strongly impact the Xco factor, an issue we examine in detail in Paper II.