A high density portion of the Orion Molecular Cloud 1 (OMC-1) contains the prominent, warm Kleinmann-Low (KL) nebula that is internally powered by an energetic event plus a farther region in which intermediate to high mass stars are forming. Its outside is affected by ultraviolet radiation from the neighboring Orion Nebula Cluster and forms the archetypical photon-dominated region (PDR) with the prominent bar feature. Its nearness makes the OMC-1 core region a touchstone for research on the dense molecular interstellar medium and PDRs. Using the Atacama Pathfinder Experiment telescope (APEX), we have imaged the line emission from the multiple transitions of several carbon monoxide (CO) isotopologues over the OMC-1 core region. Our observations employed the 2x7 pixel submillimeter CHAMP+ array to produce maps (~ 300 arcsec x 350 arcsec) of 12CO, 13CO, and C18O from mid-J transitions (J=6-5 to 8-7). We also obtained the 13CO and C18O J=3-2 images toward this region. The 12CO line emission shows a well-defined structure which is shaped and excited by a variety of phenomena, including the energetic photons from hot, massive stars in the nearby Orion Nebula's central Trapezium cluster, active high- and intermediate-mass star formation, and a past energetic event that excites the KL nebula. Radiative transfer modeling of the various isotopologic CO lines implies typical H2 densities in the OMC-1 core region of ~10^4-10^6 cm^-3 and generally elevated temperatures (~ 50-250 K). We estimate a warm gas mass in the OMC-1 core region of 86-285 solar masses.