You will use more gas to go up hill, to speed up and slow down with a full water tank. It has little or no effect once up to speed and staying at a constant speed. I usually travel with one quarter and more often a half tank. Several times a year I go somewhere that I can not fill up and run a full tank of water. (on one such trip I weighed the RV front/rear axels, using that weight to determine tire pressures).
I grew up hiking and camping in the desert. I can't imagine traveling with no water in the tank though people I respect very much do just that.