We present the ultraviolet (UV) spectroscopic evolution of a tidal disruption event (TDE) for the first time. After the discovery of the nearby TDE iPTF16fnl, we obtained a series of observations with the Space Telescope Imaging Spectrograph (STIS) onboard the Hubble Space Telescope (HST). The dominant emission features closely resemble those seen in the UV spectra of the TDE ASASSN-14li and are also similar to those of N-rich quasars. However, there is significant evolution in the shape and central wavelength of the line profiles over the course of our observations, such that at early times the lines are broad and redshifted, while at later times the lines are significantly narrower and peak near the wavelengths of their corresponding atomic transitions. Like ASASSN-14li, but unlike N-rich quasars, iPTF16fnl shows neither Mg ii λ2798Å nor C iii] λ1909Å emission features. We also present optical photometry and spectroscopy, which suggest that the complex He ii profiles observed in the optical spectra of many TDEs are in part due to the presence of N iii and C iii Wolf-Rayet features, which can potentially serve as probes of the far-UV when space-based observations are not possible. Finally, we use Swift XRT and UVOT observations to place strong limits on the X-ray emission and determine the characteristic temperature, radius, and luminosity of the emitting material. We find that iPTF16fnl is subluminous and evolves more rapidly than other optically discovered TDEs.