On our last hike, I had the chance to compare my Gore-tex shoes to my friends trail running shoes. During heavy rain having Gore-tex meant about 30 minutes worth of dry(ish) feet, after which the shoes were completely soaked. But, without the heavy rain while hiking through wetlands where the trail running shoes were soaked within minutes, Gore-tex did keep my feet relatively dry for hours.
Gore-tex seemed to be a bit more time consuming to dry, but a long evening beside a camp fire did dry them up nicely.
So in a nutshell; if it rains enough, your feet will be wet, Gore-tex or no. But, in the most typical case, where the ground is wet but it's not raining heavily, Gore-tex can keep your feet dry for hours.