We mapped the quiescent emission from the 807 GHz J = 7 --> 6 transition of CO in Orion along a strip in R.A. extending from 0.7 pc west to 1.2 pc east of theta1C Orionis. The lines arise in warm gas with temperatures greater than 40 K. The line brightness temperature is greater than 160 K in the direction of theta1C, more than twice the dust temperature, and still exceeds 35 K more than a parsec east of theta1C. The lines are narrow, with a maximum velocity width of 7 km s-1 near theta1C and decreasing to 1.5-3 km s-1 at the map boundaries. The density of the emitting gas is greater than 10(4) cm-3 and the column density exceeds 10(21) cm-2. The correlation of the bright, narrow CO(7 --> 6) lines with 158 mum [C II] emission suggests that over the entire region mapped, the narrow CO lines arise in warm photodissociation regions excited by ultraviolet (UV) photons from the Trapezium cluster. Although the Trapezium stars lie in front of the Orion A molecular cloud, not all of the warm gas is at the cloud surface. To the east of theta1C the CO(7 --> 6) lines split into two velocity components (also seen in J = 2 --> 1 (CO)-C-13 emission) which persist over several arcminutes. Since only one of these components can be on the surface, the other must arise from a dense, UV-illuminated clump or filament within the molecular cloud. Comparison of the quiescent CO(7 --> 6) emission to CO(7 --> 6) observed in a cross map of the energetic Orion KL outflow shows that the luminosity of shock-excited CO(7 --> 6) emission in Orion is only a few percent of the luminosity of the widespread quiescent CO(7 --> 6) emission.