Neil,
Edited. I had the logic of this idea wrong-way around when I first wrote this response. My apologies.
Any weight reduction would be interpreted as fuel consumption so tossing any weight would give an apparent increase in consumption. Acquiring extra weight would "game" the system but that shouldn't be a problem on a closed course.
Give every driver a few bottles of water before the weigh in.
wikipedia says gasoline weighs 6.073 lb/US gal
Gasoline - Wikipedia, the free encyclopedia
A google search gave me 6.175 lb/US gallon. It does vary with temperature - so therefore the weight of the gasoline is actually a more meaningful measure of fuel quantity than the volume.
The idea has potential, but you do need a very accurate scale of the right capacity.