Reading this achievement and the other requiring you to win a race while running out of fuel, I thought you might need to finish the race and lose while your fuel is empty. After trying this however, the solution is actually a lot simpler.
All you need to do is run out of fuel and not finish the race. The easiest way to do this is the following:
1) Waste your boosts
2) Run into obstacles on the side of the road
Doing this will quickly drain your fuel (and give you no way to dash ahead to get more). Once you've completely stopped and can't go any further, the race will end for you and this achievement will unlock.