Using the PACS and SPIRE spectrometers on board Herschel, we obtained observations of the Taffy galaxies (UGC 12914/12915) and bridge. The Taffy system is believed to be the result of a face-on collision between two gas-rich galaxies, in which the stellar disks passed through each other, but the gas was dispersed into a massive H I and molecular bridge between them. Emission is detected and mapped in both galaxies and the bridge in the [C II] 157.7 mu m and [O I]63.2 mu m fine-structure lines. Additionally, SPIRE FTS spectroscopy detects the [C I] P-3(2) -> P-3(1)(809.3 GHz) and [C I] P-3(1) -> P-3(0)(492.2 GHz) neutral carbon lines, and weakly detects high-J CO transitions in the bridge. These results indicate that the bridge is composed of a warm multi-phase medium consistent with shock and turbulent heating. Despite low star formation rates in the bridge, the [C II] emission appears to be enhanced, reaching [C II]/FIR ratios of 3.3% in parts of the bridge. Both the [C II] and [O I] lines show broad intrinsic multi-component profiles, similar to those seen in previous CO (1-0) and H I observations. The [C II] emission shares similar line profiles with both the double-peaked H I profiles and shares a high-velocity component with single-peaked CO profiles in the bridge, suggesting that the [C II] emission originates in both the neutral and molecular phases. We show that it is feasible that a combination of turbulently heated H-2 and high column-density H I, resulting from the galaxy collision, is responsible for the enhanced [C II] emission.