While I was reading a book, a book character screamed
"Shut the fuck up down there!"
I can't pick apart any phrasal verbs from this sentence.
The phrasal verb is shut up.
Shut up: To stop talking or making a noise, or to make a person or animal stop making noise:
I wish you’d shut up and listen.
Shut up! I'm trying to think.
Can you shut that dog up?
In the sentence
"Shut the fuck up down there!"
The words the fuck are most likely "semantically inert". They only serve to add emphasis. We can omit them:
Shut up down there!
The words down there reflect the position of the person(s) to whom the sentence is addressed. The person pronouncing the sentence could be on a floor that is above the addressee's, for instance.
After screaming the sentence, the book character might pick up the phone and say to his/her friend:
There are some people down there who were singing loudly. I just told them to shut up. Yes, to shut the fuck up.
Yes, Context is necessary. Out of context it parses: You (favorite colorful adjective inserted here) people need to stop making (more colorful adjectives can go here) noise (talking, singing, banging on pots, etc) Immediately.
CopperKettle's answer is almost certainly the intended meaning.
However, it could mean several things. As a shorthand, people are designated by a capital letter, such as A or B. A→B means A is talking to B.
(CopperKettle's answer.) "Shut up" means to stop talking. "The fuck" is semantically inert to add emphasis. "Down there" is a location, which could refer either to the location of the person we're talking to, or the location we expect the shutting up to occur.
A sings loudly and annoyingly while people are trying to study.
B→A: Shut the fuck up down there!
Meaning: I really want you, who is down there, to shut up, because your noise bothers me a lot.
Similar: Shut up!
As an example of the other meaning of "down there":
A is still singly loudly, but now we're outside the library.
B→A: Your singing is annoying. I'll tolerate it, but the people in the library won't. So:
B→A: Shut the fuck up down there!
Meaning: Once you get down (there) to the library, shut the fuck up!
Similar: Shut up when you're in the library!
"Shut up" is to make someone else stop talking. "The fuck" refers to a thing or (more likely) person that we are expressing contempt for. "Down there" is a location. Typically, we'd use a phrase such as "stupid fuck" or "silly fuck" instead of just "fuck" in this context, but the single word would still work.
A→B: B, you're such a fuck.
B→A: But I'm a really smart fuck!
C→A: Shut the fuck up down there!
Meaning: A, who is down there, make B, who is also down there (and is a fuck), shut up!
Similar: A, shut B up!
"Fuck up" means "mistake". "Shut down" means to close something, or make it stop functioning. "There" is a location.
A and B attempt to create a time portal, but everything goes wrong and really bad stuff is about to happen. A→B: Shut the time portal down there!
B→A: It's not a time portal, it's a fuck up!
A→B: Shut the fuck up down there!
Meaning: There's a horribly-botched attempt at making a time portal, it's near where you're standing, and I want you to shut it down.
Similar: Shut down that fucked-up excuse for a time portal!
Another alternate meaning of "there".
Meaning: Shut down the botched experiment, but do it there rather than here.
Similar: Don't shut it down here, shut it down over there!
As in (2), but now "shut up" means to literally close something. This is rare, but hypothetically correct. In this context, you'd normally see "sew up", "close up", "heal up" or "fix up" instead of "shut up".
A gets hit by a flying table-saw blade and his arm is gushing blood.
B is a doctor or nurse trained in sutures.
B→A: You fuck! I told you not to disable the safety brake!
C→B: Shut the fuck up down there!
Meaning: B (who is down there), suture the wounds of A, who is a fuck (and also down there), closed.
Similar: B, I need you to sew up A because he's a fuck!
As mentioned, (1) is probably the correct meaning, especially for something written in a book. However, (2) to (4) are, to the best of my knowledge, grammatically correct, and you're likely to hear similar things in real life. And I'm sure I missed a bunch of other interpretations. Such as being deliberately ambiguous because multiple meanings are valid. (I want you to shut that fuck up, up, then I want you to shut the fuck up too, you fucker!)
Of note, there's a lot of information missing in text that you can hear in speech, such as pauses or emphases. The first would likely be "shut the fuck up down there!" (possibly drawing out the expletive -- "shut thuuff fuck up..."); the second would be "shut the fuck, up down there!"; the third would be "shut the fuckup down, there" (the pause would be more noticeable with the alternate meaning of "there"); and the fourth would be "shut the fuck, up, down there!"
The interpretation of "Shut the fuck up down there!", depends on context.
"Fuck" could be "semantically inert" as CopperKettle said in their answer,
or "fuck up" could be a noun referring to an unknown noisemaker or some other noun, with "shut" as "lock up" or "close", and "down there" as the location to shut the fuck up in, or the current location of the fuck up which is requested to be closed.