I heard this phrase once: "He was driving slowly to save fuel." However it says about some particular fuel, which is in the tank, so using the definite article would be more logical. Why the article was omitted here?
A particular definition of 'save' is "to prevent things from being wasted", and with regard to that definition, there is no need to put an article in front of the noun as 'fuel' is being referred to generally. The same applies to money, time and effort, that is, they are all general terms that isn't bound to a single entity. However, if you really want to specifically (and unnecessarily) denote that it's your fuel you want to save, you can say something like "I was driving to save the fuel in my car."
Another reason you wouldn't use the definite article is that it has nothing to refer back to.
"I drove slowly to save the fuel."
"Oh, I mean the fuel in my car."
He's trying to save fuel as a general concept. Yes, the specific fuel is the one that is in his gas tank at the moment, but that's so obvious from context that there's no particular need to point it out; it's not like he could be saving someone else's fuel. Saying the fuel implies that we're particularly concerned about this exact, specific fuel, but we're not - fuel is interchangeable.
He was driving slowly to save fuel
This would be the typical way to express that idea. In this case, "fuel" is used as a mass noun. Note that with this sentence, fuel is still being used, but he is reducing how much (thus saving some of it).
He was driving slowly to save the fuel
This would be less likely as-is. While there is specific fuel in the tank, this would only likely be used if there had been some particular fuel referenced that was being saved:
He was driving slowly to save the fuel in the tank