This paper reports the first study of the O VI resonance line emission (1032, 1038 Angstroms) originating in the Local Bubble (or Local Hot Bubble) surrounding the solar neighborhood. In spite of the fact that O VI absorption within the Local Bubble has been observed, no resonance line emission was detected during our 230 ksec Far Ultraviolet Spectroscopic Explorer observation toward a ``shadowing filament in the southern Galactic hemisphere. As a result, tight 2 sigma upper limits are set on the intensities in the 1032 and 1038 Angstrom emission lines: 500 and 530 photons cm^{-2} s^{-1} sr^{-1}, respectively. These values place strict constraints on models and simulations. They suggest that the O VI-bearing plasma and the X-ray emissive plasma reside in distinct regions of the Local Bubble and are not mixed in a single plasma, whether in equilibrium with T ~ 10^6 K or highly overionized with T ~ 4 to 6 x 10^4 K. If the line of sight intersects multiple cool clouds within the Local Bubble, then the results also suggest that hot/cool transition zones differ from those in current simulations. With these intensity upper limits, we establish limits on the electron density, thermal pressure, pathlength, and cooling timescale of the O VI-bearing plasma in the Local Bubble. Furthermore, the intensity of O VI resonance line doublet photons originating in the Galactic thick disk and halo is determined (3500 to 4300 photons cm^{-2} s^{-1} sr^{-1}), and the electron density, thermal pressure, pathlength, and cooling timescale of its O VI-bearing plasma are calculated. The pressure in the Galactic halos O VI-bearing plasma (3100 to 3800 K cm^{-3}) agrees with model predictions for the total pressure in the thick disk/lower halo. We also report the results of searches for other emission lines.
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