Unusually strong high-excitation C I has been detected in 11 lines of sight through the Vela supernova remnant (SNR) by means of UV absorption line studies of IUE data. Most of these lines of sight lie near the western edge of the bright X-ray region of the SNR in a spatially distinct band approximately 1° by 4° oriented approximately north-south. The high-excitation C I (denoted C I* and C I**) is interpreted as evidence of a complex of shocked dense clouds interacting with the SNR, because of the high pressures indicated in this region. To further analyze the properties of this region of enhanced C I* and C I**, we present new HIRES-processed IRAS data of the entire Vela SNR. A temperature map calculated from the HIRES IRAS data, based on a two-component dust model, reveals the signature of hot dust at several locations in the SNR. The hot dust is anticorrelated spatially with X-ray emission, as would be expected for a dusty medium interacting with a shock wave. The regions of hot dust are strongly correlated with optical filaments, supporting a scenario of dense clouds interior to the SNR that have been shocked and are now cooling behind the supernova blast wave. With few exceptions, the lines of sight to the strong high-excitation C I pass through regions of hot dust and optical filaments. Possible mechanisms for the production of the unexpectedly large columns of high-excitation C I are discussed. Dense clouds on the back western hemisphere of the remnant may explain the relatively low X-ray emission in the western portion of the Vela SNR due to the slower forward shock velocity in regions where the shock has encountered the dense clouds. An alternate explanation for the presence of ground-state and excited-state neutrals, as well as ionized species, along the same line of sight is a magnetic precursor that heats and compresses the gas ahead of the shock.