+1The answer is Yes
It affects mileage because you are allowing more air flow under the truck, which equals drag. If you notice the stock FJ has a Rake to the front end. Even though it doesn't look as good as level, the design helps mileage. It may not be a drastic change, but it will affect it.
I don't think it's the weight of the tire so much as it is the rolling resistance of larger tires plus the effect on gearing. Larger tire means lower gear ratio. I found that out on my 5 speed manual 2500HD Silverado crew cab. A few pounds in tire weight meant nothing to a 6500 pound truck. It was the extra circumference and rolling resitance that affected the mileage by loading the engine slightly more than the stock tire.+1
In addition, most folks folks put on larger tires that weigh more than stock after a lift, and the unsprung weight of heavier tires is really hard on fuel economy because you continually have to spin that extra weight every time you touch the gas pedal.
Another YES from me. Any mod which adds weight or changes aerodynamics will decrease mileage even if only slightly: heavy skids, heavy tires, steel wheels, heavy bumpers/winch/extra battery, aftermarket rack, lift kit, hi-lift jack, all the extra crap you carry in the back.
If you've already installed off-road tires and encountered the loss of mileage that comes with them you may not notice the additional decrease in mileage.
Although I still wish that I could get better mileage, I do feel somewhat fortunate that, even with all of my mods, I am able to average 16-17 mpg with careful driving on my interstate only trips back and forth to Colorado/Utah.