Can the manufacturer order it be removed from from an operating aircraft that’s scheduled to fly, or just one that’s spare?
It depends. As wiedehopf notes, some operators depend on lease engines for day-to-day operations, others only use them to fill in when something unexpected happens.
An example of the latter might be an operator had to pull a couple engines unexpectedly - and it'll be several months before they come back out of overhaul. So they lease an engine for six months to tide them over, figuring at the end of six months they'll be good again. Except that at the end of that six months, they've had a couple more engines go south unexpectedly and they still need the lease engine. They go back to the engine manufacture and ask to extend the lease - at that point the manufacture can legitimately tell them no, I need the engine back, I've promised it to XYZ - sorry... Of course if they don't need it back they'd be happy to extend the lease.
I used to sit in technical reviews with all three major manufactures - and one of the first things on the agenda was spares status - both of the engines owned by the operators and the manufacture's spares pool. Usually it was routine, but occasionally it would be tight and there was concern that an unplanned removal could force an AOG situation... However I can't recall an actual extended AOG situation caused by lack of spare engines (beyond the 'normal' time to get a crew and spare to remote airport and R&R the engine).