Infrared dark clouds (IRDCs) are dense, molecular structures in the interstellar medium that can harbour sites of high-mass star formation. IRDCs contain supersonic turbulence, which is expected to generate shocks that locally heat pockets of gas within the clouds. We present observations of the CO J = 8-7, 9-8, and 10-9 transitions, taken with the Herschel Space Observatory, towards four dense, starless clumps within IRDCs (C1 in G028.37+00.07, F1 and F2 in G034.43+0007, and G2 in G034.77-0.55). We detect the CO J = 8-7 and 9-8 transitions towards three of the clumps (C1, F1, and F2) at intensity levels greater than expected from photodissociation region (PDR) models. The average ratio of the 8-7 to 9-8 lines is also found to be between 1.6 and 2.6 in the three clumps with detections, significantly smaller than expected from PDR models. These low line ratios and large line intensities strongly suggest that the C1, F1, and F2 clumps contain a hot gas component not accounted for by standard PDR models. Such a hot gas component could be generated by turbulence dissipating in low velocity shocks. ; Comment: 14 pages, 8 figures, 5 tables, accepted by A&A, minor updates to match the final published version