"adding fuel to the fire"
Should there be no articles in this idiom? Can someone give an explanation as to why there is one?
Idioms are generally fixed phrases and thus sometimes use phrasing we wouldn't otherwise use, possibly due to preserving a way of speaking/writing that has disappeared in other contexts.
In this case, the idiom "adding fuel to the fire" means someone is doing something to make a situation worse, so when you use it there is generally a specific situation being referred to, so it would make sense to see the article here.
With, "add insult to injury" apart from the fixed phrase aspect, there's also the possibility of an archaic meaning. The idiom is quite old (being from a translation of a Latin story), but "injury" also used to mean "injustice". We can speak of an injustice, but you can also speak of insult and injustice/injury as states.
I suffered injustice
I endured insult
Thus, "adding insult to injury" can be to add the state of being insulted to the state of being wronged (either an injustice or an injury in the modern sense)
This is an interesting question (especially in light of the "insult to injury" comment). I'm not sure I'll be able to explain why the expressions are the way they are, but I will try to explain how I think they would change. I will explain them literally and them abstract them.
Adding fuel to the fire vs adding fuel to fire.
"Adding fuel to the fire" sounds like there is a specific fire somewhere that you are adding fuel to. In the abstract sense, it sounds like there is a specific argument going on that you are provoking or making worse.
"Adding fuel to fire" sounds like the speaker is about to talk about the actual act of "adding fuel to fire". It sounds like it will be followed by a warning or a statement of how. In the abstract sense, it gives the impression that there is not a specific argument taking place (note the emphasis on not - omitting the article sounds very deliberate) and that you are talking about something that would make any argument, not just one about a specific topic, worse.
Adding insult to the injury versus adding insult to injury.
This one is harder and I would believe if someone told me that the original expression was "Adding insult to an/the injury" and the article was dropped over time. For this one, my only thought is that the expression itself is somewhat abstract so it requires generalizing of the nouns. By abstract, I mean, both "insult" and "injury" are intangible concepts rather than physical things that can be seen. (Side note: for this expression to apply properly, the "injury" must be emotional and not physical.)
When I hear, adding insult to the injury", I get the strong impression that the injury is a physical wound rather than an emotional thing. And once that impression is given, the idea of adding an insult (again, non physical) to a wound sounds odd.
Something has gone wrong, that's for sure! There's an unwanted fire. Thus we don't want to add fuel to the fire (the unwanted one). We might however toss more wood onto the fire in the living room.
And there's no article in the idiom "add insult to injury"
Whilst I can imagine someone making the situation worse by adding fuel to an unwanted fire, it is hard to imagine someone insulting an injury. Let's say you injure yourself by hitting your thumb with a hammer. The injury is not going to get worse if you start yelling insults at it.
However it is believable that you might be injured (ie. punched) and then insulted. Thus in that situation your attacker "added" insult to injury. (However they didn't insult the injury).
In fact, the article is not necessarily required. It depends on the usage. A sentence like the following is possible, where the article is glibly elided:
"Adding fuel to fire, Bob called Jack's marketing plan imbecilic."
There are various situations and stylistic nuances in English which allow articles to be elided. For instance newspaper headlines, or phrases like "boy meets girl", or when embedded clauses modify a word, as in, "The entire movie was yet another reluctant-hero-with-father-issues-saves-world cliché".
When the article is present, which is usually the case, it is "the" and not "a": "adding fuel to the fire", not "adding fuel to a fire".
The reason is that the expression build a metaphor: the volatile situation is imagined to be a fire. One fire. That fire is that specific fire to which metaphoric "fuel" is added (whatever provocation that is causing the situation to escalate).
"adding fuel to a fire" evokes a meaning revolving around searching the world for some fire somewhere and then adding fuel to it; not necessarily the fire that serves as the metaphor for the situation.
Suppose the idea were expressed without a metaphor, by saying something like "Bob made the situation worse by calling Jack's marketing plan imbecillic."
It has to be "the situation" and not "a situation". Since the fire represents the situation, it follows the same way: "the fire" and not "a fire".
The 'the' removes an ambiguity between 'fire' as a noun and 'fire' as a verb.
He was adding fuel to fire the steam engine.
He was adding fuel to the fire.
The idiom is based on the use of 'fire' as a noun, so is taken from the second usage. 'Injury' isn't a verb, so there is no need to remove ambiguity.
It may be due to phrasiology.
"He's adding fuel to the fire" / "You're adding fuel to the fire" - you would usually use that phrase to talk about a specific person worsening a specific situation.
"Don't add insult to injury" by contrast, is usually issued as direct advice.
However - you can say "You're adding insult to injury", that isn't abnormal.
So I don't fully buy your premise that a definite article in unheard of - and if it is more common in this case it is because of how it is commonly phrased.