With an aim of probing the physical conditions and excitation mechanisms of warm molecular gas in individual star-forming regions, we performed Herschel SPIRE FTS observations of 30 Doradus in the LMC. In our FTS observations, important FIR cooling lines in the ISM, including CO J=4-3 to 13-12, [CI] 370 micron, and [NII] 205 micron, were clearly detected. In combination with ground-based CO data, we then constructed CO spectral line energy distributions (SLEDs) on 10 pc scales over a 60 pc x 60 pc area and found that the shape of the observed CO SLEDs considerably changes across 30 Doradus, e.g., the peak transition varies from J=6-5 to 10-9, while the slope characterized by the high-to-intermediate J ratio ranges from 0.4 to 1.8. To examine the source(s) of these variations in CO transitions, we analyzed the CO observations, along with [CII] 158 micron, [CI] 370 micron, [OI] 145 micron, H2 0-0 S(3), and FIR luminosity data, using state-of-the-art models of PDRs and shocks. Our detailed modeling showed that the observed CO emission likely originates from highly-compressed (thermal pressure ~ 1e7-1e9 K cm-3) clumps on 0.7-2 pc scales, which could be produced by either UV photons (UV radiation field ~ 1e3-1e5 Mathis fields) or low-velocity C-type shocks (pre-shock medium density ~ 1e4-1e6 cm-3 and shock velocity ~ 5-10 km s-1). Considering the stellar content in 30 Doradus, however, we tentatively excluded the stellar origin of CO excitation and concluded that low-velocity shocks driven by kpc scale processes (e.g., interaction between the Milky Way and the Magellanic Clouds) are likely the dominant source of heating for CO. The shocked CO-bright medium was then found to be warm (temperature ~ 100-500 K) and surrounded by a UV-regulated low pressure component (a few (1e4-1e5) K cm-3) that is bright in [CII] 158 micron, [CI] 370 micron, [OI] 145 micron, and FIR dust continuum emission.