We use hydrodynamic simulations to predict correlations between Lya forest absorption and galaxies at redshift z~3. The probability distribution function (PDF) of Lya flux decrements shifts systematically towards higher values in the vicinity of galaxies, reflecting the overdense environments in which these galaxies reside. The predicted signal remains strong in spectra smoothed over 50-200 km/s, allowing tests with moderate resolution quasar spectra. The strong bias of high redshift galaxies towards high density regions imprints a clear signature on the flux PDF, but the predictions are not sensitive to galaxy baryon mass or star formation rate, and they are similar for galaxies and for dark matter halos. The dependence of the flux PDF on galaxy proximity is sensitive to redshift determination errors, with rms errors of 150-300 km/s substantially weakening the predicted trends. On larger scales, the mean galaxy overdensity in a cube of 5 or 10 Mpc/h (comoving) is strongly correlated with the mean Lya flux decrement on a line of sight through the cube center. The slope of the correlation is ~3 times steeper for galaxies than for dark matter as a result of galaxy bias. The predicted large scale correlation is in qualitative agreement with recently reported observational results. However, observations also show a drop in absorption in the immediate vicinity of galaxies, which our models do not predict even if we allow the galaxies or AGNs within them to be ionizing sources. This decreased absorption could be a signature of galaxy feedback on the surrounding IGM, perhaps via galactic winds. Peculiar velocities often allow gas at comoving distances ~1.5 Mpc/h to produce saturated absorption at the galaxy redshift, so any feedback mechanism must suppress neutral hydrogen out to these radii to match the data. (Abridged)
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