We report the first astronomical detection of isotopic 13CO J = 6 → 5 emission at 661 GHz from the molecular clouds in Orion A and in NGC 2024. Strong lines (Rayleigh-Jeans main-beam brightness temperature ≈ 50 K) indicate large column densities of warm, dense, molecular gas in these star formation regions. From the new 13CO data, together with complementary 13CO J = 3 → 2 and 12CO J = 6 → 5 measurements, we estimate that H2 column densities of gas with temperature ≥ 100 K lie between 3 × 1022 and 4 × 1023 cm-2. These column densities, corresponding to ≥30% of the total molecular gas mass in the observed star-forming regions, are difficult to explain by present theoretical heating models. In NGC 2024 and the Orion Bar, where lines are narrow (ΔυLSR ≤ 3 km s-1), shock heating is very unlikely. The correlation of warm CO column density with UV intensity, also indicated by the nondetection (Tmb < 10 K) of 13CO J = 6 → 5 emission toward the weak UV source NGC 2023, favors heating by UV radiation. However, present photodissociation region (PDR) models fail to predict the large quantities of warm gas detected in NGC 2024 and in the Orion Bar.