For Orthodox Jews, Sabbath observance includes abstaining from certain kinds of "work" from sundown Friday to nightfall Saturday. Work includes things like writing, using electricity, etc. These prohibitions are overridden by the moral imperative to save a life, but of course much of the work done by residents isn't lifesaving.
Many Orthodox Jewish students try to find residencies which will allow them to not work on the Sabbath. Some residency programs have official places for one or two residents who are Sabbath observant. Some who are generally pretty flexible with their residents' schedules will accommodate a Sabbath-observant resident. This may sometimes include having the resident swap 2 for 1 with someone else to avoid Saturday call, etc. And of course many programs simply can't or won't accommodate it.
Think Maimonides in Brooklyn may as well. But not 100% sure on that one. Believe some concessions were made out in long island as well - NUMC, I believe. Good luck!