Well, they can’t all be winners I suppose and this Puzzler definitely goes in the loss column. The given answer, that the truck would burn enough fuel to get under the weight specification, is as petty as they come.
First of all, there isn’t a truck scale in the world that is going to have the necessary precision or accuracy to account for the weight of a sparrow. A typical truck scale is only precise to divisions of 20 to 50 pounds. There is no way, given any reasonable technology, to know that the truck weighs “exactly 20 tons” even if the unit was calibrated moments before.
Furthermore, the load of the truck on the bridge is going to vary continuously as it bumps along the bridge. These differences in load are insignificant in the context of the capacity of the bridge, but will far exceed the weight of a sparrow (~ 1 oz.) or even the weight of the fuel expended (no more than a few pounds).
Lastly, unless the folks running the show in Borneo are idiots, they would certainly be applying a safety factor when determining the capacity of the bridge to account for these points (and others).
In conclusion, the answer is right, but for the wrong reason. The weight of the fuel or the sparrow are utterly insignificant. The driver doesn’t need to be concerned with any additional load on the bridge unless a Borneo elephant somehow hitches a ride.