With an aim of probing the physical conditions and excitation mechanisms of warm molecular gas in individual star-forming regions, we performed Herschel SPIRE Fourier Transform Spectrometer (FTS) observations of 30 Doradus in the Large Magellanic Cloud. In our FTS observations, important far-infrared (FIR) cooling lines in the interstellar medium, including CO J = 4–3 to J = 13–12, [C I] 370 μm, and [N II] 205 μm, were clearly detected. In combination with ground-based CO J = 1–0 and J = 3–2 data, we then constructed CO spectral line energy distributions (SLEDs) on ~10 pc scales over a ~60 pc × 60 pc area and found that the shape of the observed CO SLEDs considerably changes across 30 Doradus. For example, the peak transition Jp varies from J = 6–5 to J = 10–9, while the slope characterized by the high-to-intermediate J ratio α 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 [C II] 158 μm, [C I] 370 μm, [O I] 145 μm, H2 0–0 S(3), and FIR luminosity data, using state-of-the-art models of photodissociation regions and shocks. Our detailed modeling showed that the observed CO emission likely originates from highly compressed (thermal pressure P∕kB ~ 107–109 K cm−3) clumps on ~0.7–2 pc scales, which could be produced by either ultraviolet (UV) photons (UV radiation field GUV ~ 103–105 Mathis fields) or low-velocity C-type shocks (pre-shock medium density npre ~ 104–106 cm−3 and shock velocity vs ~ 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 kiloparsec-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 T ~ 100–500 K) and surrounded by a UV-regulated low-pressure component (P∕kB ~ a few (104 –105) K cm−3) that is bright in [C II] 158 μm, [C I] 370 μm, [O I] 145 μm, and FIR dust continuum emission.