We present a transmission spectrum for the warm (500-600 K) sub-Neptune HD 3167c obtained using the Hubble Space Telescope Wide Field Camera 3 infrared spectrograph. We combine these data, which span the 1.125-1.643 μm wavelength range, with broadband transit measurements made using Kepler/K2 (0.6-0.9 μm) and Spitzer/IRAC (4-5 μm). We find evidence for absorption by at least one of H2O, HCN, CO2, and CH4 (Bayes factor 7.4; 2.5σ significance), although the data precision does not allow us to unambiguously discriminate between these molecules. The transmission spectrum rules out cloud-free hydrogen-dominated atmospheres with metallicities ≤100 solar at >5.8σ confidence. In contrast, good agreement with the data is obtained for cloud-free models assuming metallicities >700 solar. However, for retrieval analyses that include the effect of clouds, a much broader range of metallicities (including subsolar) is consistent with the data, due to the degeneracy with cloud-top pressure. Self-consistent chemistry models that account for photochemistry and vertical mixing are presented for the atmosphere of HD 3167c. The predictions of these models are broadly consistent with our abundance constraints, although this is primarily due to the large uncertainties on the latter. Interior structure models suggest that the core mass fraction is >40%, independent of a rock or water core composition, and independent of atmospheric envelope metallicity up to 1000 solar. We also report abundance measurements for 15 elements in the host star, showing that it has a very nearly solar composition.