Strong C III λ977 and O VI λλ1032, 1038 emission lines are detected in Far Ultraviolet Spectroscopic Explorer (FUSE) spectra of N49, the brightest optical supernova remnant in the Large Magellanic Cloud. Shocks with velocities ranging from less than 130 to more than 180 km/s are responsible for these emission lines and are present over the entire eastern half of the remnant. The emission lines are very broad, spanning about 700 km/s in all the spectra. The ratio of C III to O VI at different velocities shows that fast-moving radiative shocks are present in N49. These may be a result of instabilities, secondary shocks, and turbulent mixing. The velocity structure of O VI emission from a 4''x20'' region along the southeast edge of the remnant is the same as that from a larger 30''x30'' region, which is evidence for equally complex structure at multiple scales. An archival HST WFPC2 Hα image also shows highly filamentary morphology, with structures on subarcsecond scales. The complexity of the emission suggests that the effective interstellar extinction at far-ultraviolet wavelengths is smaller than that at optical wavelengths. The uncertainty in the far-ultraviolet extinction translates to an order of magnitude uncertainty in the intrinsic O VI intensity of N49. A comparison of the O VI 1032 and 1038 Å line profiles shows that the latter is affected by absorption due to molecular hydrogen, probably associated with the remnant and its surroundings, as well as due to O I in an intermediate-velocity cloud in the Galactic halo.