move occurs because value has type `RefCell`, which does not implement the `Copy` trait

into_inner() method will consume the RefCell instance to give you the wrapped value.

To achieve that, it requires you to have ownership over the instance of RefCell. But you don't have that as it is within Rc and unless you consume the Rc as well to get ownership of RefCell, you can't call into_inner().

In your code, due to deref coersion, you get immutable ref to the inside RefCell so you can only call methods that accept &self.

If you want mutate the contents inside RefCell, you can do it as follows:

#![allow(dead_code)]
#![allow(unused_variables)]

use std::rc::Rc;
use std::cell::RefCell;

struct InstanceContext {
    id: i32,
    instances: Option<Vec<String>>,
}

fn main() {
    let instance_context = Rc::new(RefCell::new(InstanceContext { id: 5, instances: None }));
    instance_context.borrow_mut().instances = Some(vec![String::from("abc")]);
}

Playground

borrow_mut() takes an immutable ref to RefCell instance and lets you get a mutable ref to the wrapped value with which you can mutate the contents of the wrapped value.