Actually, just the opposite happens. Bigger tires/wheels get less fuel milage. It is unsprung weight. Not only do big big tires/wheels destroy vehicle dynamics, but mileage goes down as well. Difference between 18 and 19, no likely not so much. However, between 18 and 22, without question.
With the bigger tire, the engine has to sling more outside weight around from every stop. Imagine inertia, and holding on to a spinning bike tire by the axle. The forces acting on it are amazing. Applying the power, actually torque to spin that big ole dub from a stop, or simply applying any acceleration will take more gas. Not to mention more wear on all downstream components. . .such as brakes, bearings, hubs, shock towers, etc.
I have taken my LR3 HSE with 19's offroad many times. However, I am not doing any true rubicon rock crawl stuff. The 19's are fine for typical and consisten off road use. I have tossed around the idea of having a purpose set of 18" wheels and off road tires that I could swap out however.