We map for the first time the two-dimensional H_2 excitation of warm intergalactic gas in Stephan's Quintet on group-wide (50 × 35 kpc^2) scales to quantify the temperature, mass, and warm H_2 mass fraction as a function of position using Spitzer. Molecular gas temperatures are seen to rise (to T > 700 K) and the slope of the power-law density–temperature relation flattens along the main ridge of the filament, defining the region of maximum heating. We also performed MHD modeling of the excitation properties of the warm gas, to map the velocity structure and energy deposition rate of slow and fast molecular shocks. Slow magnetic shocks were required to explain the power radiated from the lowest-lying rotational states of H_2, and strongly support the idea that energy cascades down to small scales and low velocities from the fast collision of NGC 7318b with group-wide gas. The highest levels of heating of the warm H_2 are strongly correlated with the large-scale stirring of the medium as measured by [C ii] spectroscopy with Herschel. H_2 is also seen associated with a separate bridge that extends toward the Seyfert nucleus in NGC 7319, from both Spitzer and CARMA CO observations. This opens up the possibility that both galaxy collisions and outflows from active galactic nuclei can turbulently heat gas on large scales in compact groups. The observations provide a laboratory for studying the effects of turbulent energy dissipation on group-wide scales, which may provide clues about the heating and cooling of gas at high z in early galaxy and protogalaxy formation.