Without having to carry any extra items and thus extra weight, this is what you can do to dry your boots, Gore-tex or otherwise, in the field, in above-freezing temperatures:
Prevention: Make every effort to keep your feet dry in the first place.
Sometimes it's just inevitable though.
Absorb excess water: After removing your insoles, use a highly absorbent synthetic shammy (like those for drying cars) to pat down and dry the footwear by hand as much as possible, wringing out the shammy frequently. A typical camp towel will work, but not nearly as effectively. Repeat until no more water is absorbed into the shammy. Likewise, take your wet socks, roll them up in the shammy and wring the whole thing out. Repeat until no more water wrings out.
Controlled application of heat: If still hiking, return the boots with insoles back to your feet and continue on. The heat from your feet will do a lot to dry your boots, but only if you've removed most of the water by absorbing it. Don't put fresh socks on yet. (Obviously if the boots still will be getting wet then don't bother drying them until you can get to camp.) When at camp, after absorbing the majority of the water, wear the boots around while setting up camp, cooking, etc. so that your body heat can continue to work on moving the moisture out. I'd recommend dry socks for this part, but no socks could also work if it's warm enough.
Another method to apply heat is to warm up some small rocks by the fire and then place them in your boots, making sure that they are not too hot, otherwise you'll ruin your boots. You should be able to touch them without getting burned. Periodically replace the rocks with more warm ones. A similar, possibly superior, method is to take two water bottles, fill them with near-boiling water and place one in each boot. (This is better suited to overnight drying.) The drawback with the water bottles is that you likely will use valuable fuel to heat that water, unless you can heat the water with the fire.
Finally, when going to sleep, put some freshly warmed stones/water bottles in your mostly dry/barely damp boots, place the boots in a stuff sack or two, then put the boots in the foot of your sleeping bag with you. This will keep them near heat all night. Although it won't be the most comfortable thing in the world, it's not all that bad. However, don't do this if your boots are still quite saturated. If that's the case, just leave them inside the tent/tarp.
I do not recommend drying boots directly by the fire. It's very difficult to gauge and control how much heat is being applied and you'd greatly risk destroying your boots. Too much heat will shrink and crack leather, melt synthetic materials, and melt the glue holding your sole to the boot.
In sub-freezing temps, the procedure is similar, but wearing boots to dry them is not the ideal option. Using the hot water bottle/stone method is the way to go then. I'd also note that having the boots near a fire is good, just not closer than you'd want to sit yourself. If it won't burn you, it won't burn your boots.
And one final thought: sometimes there are conditions where you just have to settle for various degrees of wetness/dryness in your footwear, making sure that at least your feet are dry when you sleep.
