Why is the definite article used in: ”What happened to the rest of "the" soup”?

The OP is right that the phrase 'the rest of the soup that we ate yesterday', taken literally, is absurd. The problem, however, is not a matter of the use of the definite article, nor of grammar at all. We need the definite article here because we are trying to refer to some definite soup.

The problem us that we are trying to refer to the whole of the soup by a characteristic (being eaten yesterday) that is true of only a part of it, but we are doing it with a wording ('the . . . that . . .') that implies that the characteristic is true of the whole of it. We thus, if our words are taken literally, end up inadvertently referring to the wrong thing, the soup that has already been consumed, and this creates an absurdity when combined with the rest of.

The soup that we want to refer by the soup that we ate yesterday is not the soup that we ate yesterday, but the soup some of which we ate yesterday. If we wanted to be really, really pedantic, as people are in, say, drafting legal documents, we would say something like that. For the purposes of everyday communication, we can take it for granted that the sentence will be understood as if it contained some of which, even though it doesn't.

Your confusion is coming from determining what counts as the soup. Soup is something created in quantity, so there is often the case of some of it being left over. When people ask about the rest of something, they are referring to what was left over, that is, unconsumed and still existing. That the question is being asked means that someone moved the leftover soup somewhere unknown. You need the here because you were talking about a very particular leftover soup, and the adjective phrase underscores that point.

The title of your question suggests it's about why the definite article is used, and the first part of your question why a definite article as opposed to an indefinite article (or no article). In terms of this particular sentence, it needs a definite article because it's referring to a specific soup—that is, the soup we ate (a portion of) yesterday.

At first glance this appears to perhaps be more generally related to the phrase the rest of X. The phrase seems to lend itself to objects which, if taking an article, would use the definite one. But exceptions readily come to mind: for example, "What happens to the rest of an afternoon [once you've done xyz with it]?" (Incidentally, it's difficult to think of a context in which one would use an indefinite article after the past tense happened to... perhaps that's also why the example sentence demands a definite article? But that's probably not the point the author of the Learner's Guide was making.)

But your more urgent point seems to be not so much the in/definiteness of the article, but rather what noun phrase is properly the object of the question. And rightly so, since, as you point out, the sentence could be parsed in a way that makes the question border on the absurd.

If we start with the basic form of the question, "What happened to X?" X is made up of three elements: the rest [of], the soup, and that we ate yesterday. One way to parse this sentence is with the soup as the main noun,† which we'll call β. So you start with β (the soup) and modify it with the other two—which you can do in either order, interestingly. The soup can become the rest of the soup or the soup that was eaten yesterday, but either way when you modify it with both you end up with "the rest of β that was eaten yesterday."

In other words, if the soup is the main noun of the sentence, the question could be rephrased as something like: "What happened to the remainder of that portion of soup we ate [yesterday]?" Or, as you put it, 'What happened to the (rest of the) soup that's now in our stomachs?" To which the answer might be, "Ummm...well, there isn't any left to speak of, if you're asking about the portion we already ate." Parsed this way, the question is logically its own answer: there was some soup; yesterday we ate a portion of it. What happened to the portion we ate? We ate it—what is there to inquire about? There is no remainder of the (portion of) soup that we ate yesterday. Or to go back to the β formulation: there can't be a remainder of some thing, β, that was eaten yesterday.

Of course, your interlocutor would assume you weren't asking this version of the question since you're likely attempting to engage in meaningful interaction.‡ Given the assumption that the purpose of language is meaningful exchange, the main noun of the sentence must be the rest. (It's mind boggling to think that this calculation is something we do as interlocutors almost instantaneously, yet attempting to reason out and describe why the calculation is logical has taken me hours of pondering). In this case, of the soup modifies the rest and that we ate yesterday modifies the soup. So in the basic form "What happened to X?" X is the rest, so that what we're asking is what happened to the remainder [of the soup—that soup we ate yesterday]?

†I believe this is referred to in functional linguistics as the head of the nominal group, but I'm not a linguist and I could be anywhere from slightly wrong to completely off base.

‡This is reminiscent of Grice's Maxims. None of the maxims is strictly applicable here, but underlying all of them is the rationale that communication is a cooperative effort aimed at meaningful exchange. Thus, assuming we're not characters in a Samuel Beckett play, we presume our interlocutors aren't asking questions which are, on their face, absurd.

Having been encouraged to re-post my comment (which has no references I'm afraid) as an answer here it is:

You have provided the correct answer to this yourself when you wrote "There was soup. We ate some of it and left the rest over. What happened to the rest of the (whole) soup" the idea is that a batch of soup was made and we ate (or drank or had) some of it. The question is then about the whole batch.

Perhaps it makes more sense when talking about solids, for example "What happened to the rest of the turkey we ate yesterday?" There is usually leftover turkey, they're huge.

Having said that I think 'had' is more common than 'ate' in this context and avoids the logical conundrum.