Here is as far as I could get, using rental, partly based on How can I store a Chars iterator in the same struct as the String it is iterating on?. The difference here is that the get_iter method of the locked member has to take a mutable self reference.
I'm not tied to using rental: I'd be just as happy with a solution using reffers or owning_ref.
The PhantomData is present here just so that MyIter bears the normal lifetime relationship to MyIterable, the thing being iterated over.
I also tried changing #[rental] to #[rental(deref_mut_suffix)] and changing the return type of MyIterable.get_iter to Box<Iterator<Item=i32> + 'a> but that gave me other lifetime errors originating in the macro that I was unable to decipher.
#[macro_use]
extern crate rental;
use std::marker::PhantomData;
pub struct MyIterable {}
impl MyIterable {
    // In the real use-case I can't remove the 'mut'.
    pub fn get_iter<'a>(&'a mut self) -> MyIter<'a> {
        MyIter {
            marker: PhantomData,
        }
    }
}
pub struct MyIter<'a> {
    marker: PhantomData<&'a MyIterable>,
}
impl<'a> Iterator for MyIter<'a> {
    type Item = i32;
    fn next(&mut self) -> Option<i32> {
        Some(42)
    }
}
use std::sync::Mutex;
rental! {
    mod locking_iter {
        pub use super::{MyIterable, MyIter};
        use std::sync::MutexGuard;
        #[rental]
        pub struct LockingIter<'a> {
            guard: MutexGuard<'a, MyIterable>,
            iter: MyIter<'guard>,
        }
    }
}
use locking_iter::LockingIter;
impl<'a> Iterator for LockingIter<'a> {
    type Item = i32;
    #[inline]
    fn next(&mut self) -> Option<Self::Item> {
        self.rent_mut(|iter| iter.next())
    }
}
struct Access {
    shared: Mutex<MyIterable>,
}
impl Access {
    pub fn get_iter<'a>(&'a self) -> Box<Iterator<Item = i32> + 'a> {
        Box::new(LockingIter::new(self.shared.lock().unwrap(), |mi| {
            mi.get_iter()
        }))
    }
}
fn main() {
    let access = Access {
        shared: Mutex::new(MyIterable {}),
    };
    let iter = access.get_iter();
    let contents: Vec<i32> = iter.take(2).collect();
    println!("contents: {:?}", contents);
}
 
    