It does. Fuel is burnt according to engine conditions.
But, as in real life, you'd have a hard time seeing the gauge move unless you let it idle for several hours. And I think LFS now turns off your engine after a minute or two to save network traffic anyway.
You can also save fuel by short shifting, or using less throttle, which is useful if you ever miss the pitlane entrance in the thick of racing and need to squeeze another lap in on less than normal fuel.