I think it depends on the specialty.
Let's look at the typical IM service in the hospital - given that's what I know best. A team is usually structured as 1 attending, 1 senior, 1-2 interns, some number of medical students. There's lots of variants on this same structure, but I think that's most common. They can see (by ACGME mandate) a maximum of 16 (for 1 intern teams) to 20 patients (for 2 interns). Now, those are caps - I'd say a typical team probably averages 13-17 patients, with some days being more and others being less. Usually, but not always, the attending is *exclusively* on that inpatient service when he's on service. Some places do have an attending cover more than 1 team (particularly for weekends/holidays) or they may have a more traditional GIM practice where they have their own clinic schedule as well, but usually it's just 1 attending, 1 team, no other responsibilities.
What would happen if all the residents disappeared for a day? Well... you'd be left with the attending and 13-17 patients. Guess how many patients a community hospitalist sees in a given workday by themselves? Hint: The average is usually around 15.
So if the residents disappeared for a day or two, the daytime inpatient medicine services would function just fine. The attendings would be stuck doing all of the work themselves - but they'd also have no education to do (for obvious reasons). Now, their schedules would be all kinds of messed up - academic attendings tend to have radically different schedules than community hospitalists - namely that they *typically* take fewer days off due to the less intense nature of their work (though likely more weekends). But it would work just fine. And the staff would be miserable. But the residents are only "necessary" to give the attendings the relatively cush life that allows them to accept the smaller salary (and lets them take time for teaching, QI, whatever else they like to do).
Nights gets a bit hinky, because most places just have residents operating semi-autonomously at night with home attending backup. So for facilities that don't have a night in house attending, they'd have to screw around with the schedules even more - but again, the attendings should be perfectly capable of triaging night calls and admissions on their own. At least for a few days before they all up and quit.
What about consult services? Same story, though there the schedules would be even more screwed up. The residents allow the subspecialists to likely see more patients than they otherwise would - or spend less time doing consultations and more time doing whatever procedures/clinic/whatever they would rather be doing.
The ICU would be a bit more touch and go, particularly at night - but even at the program where I did residency that had ridiculous ICU volume, I think the actual attendings could manage on their own for a few days before they all quit.
Clinics get more of a questionmark, because an attending is often supervising 4 individual residents with their individual schedules. Now, none of them is as busy as a full attending on their own - but even if they were only a third as busy as an attending, that's still 1.3X a clinic schedule.
I think it would be a lot less tenable for someone like a surgeon if the residents disappeared - because there's often no other built in mechanism for first assists and so much of the grunt work done by the residents saves the surgeons time and allows them to increase case volumes.
Basically, I'd say that most academic hospital services *could* function *briefly* if the residents up and disappeared, but that would require the attendings to work at full clinical volume on their own and that could not be sustainable due to the overall staffing not being built for that.