Gittin 8b: Although it is prohibited to ask a non-Jew to do melachah on Shabbos, here for the sake of Yishuv Eretz Yisroel the Sages relaxed their decree.
Tosafos: But for any other mitzvah it is not allowed to ask a non-Jew to perform a Torah prohibition….
Certainly for the sake of Bris Milah it is allow to ask him to perform a Rabbinically prohibited act… but one cannot learn from here that it is allowed to ask a non-Jew to carry a sefer on Shabbos through a Rabbinically prohibited street, because only for Bris Milah, which itself pushes aside Shabbos, were they lenient.
Shulchan Aruch 307:5 quoting Rambam: It is permitted to ask a non-Jew to do an act of work that is Rabbinically prohibited, for the needs of a sick person, a great need (large loss) or a mitzvah. And some (Tosafos) disagree with this.
גיטין ח: אע”ג דאמירה לעובד כוכבים שבת משום ישוב ארץ ישראל לא גזרו רבנן. תוס’: אבל משום מצוה אחרת לא היינו מתירין אמירה לעובד כוכבים במלאכה דאוריית… ודאי איסורא דרבנן שרי בחצר שלא עירבו משום מצות מילה… ואין ללמוד מכאן היתר לומר לעובד כוכבים להביא ספר בשבת דרך כרמלית דלא דמי דדוקא משום מילה דהיא גופה דחיא שבת התירו.
שו”ע או”ח ש”ז ס”ה: דבר שאינו מלאכה ואינו אסור לעשותו בשבת אלא משום שבות מותר לישראל לומר לא״י לעשותו בשבת והוא שיהיה שם מקצת חולי או יהיה צריך לדבר צורך הרבה או מפני מצוה כיצד אומר ישראל לא״י בשבת לעלות באילן להביא שופר לתקוע תקיעת מצוה או להביא מים דרך חצר שלא עירבו לרחוץ בו המצטער ויש אוסרין.
One Friday night when Rabbi Yisroel Reisman was recently married and had just gotten semicha, his landlord asked him a shailah. He had the cholent fully cooked before Shabbos and his practice was to set the crockpot on a timer so that it would be off before the daytime meal when he removed the food from the pot. (Possibly he did this to avoid the prohibition of dishing food out from a pot that is currently standing on the fire – see Orach Chaim 318:18.) The problem was that just after Shabbos began, he realized that he had plugged the crockpot into the timer but forgotten to plug the timer into the wall. Could he call a goy to move the pot onto the blech?
Rabbi Reisman showed him that at the end of Siman 253, the Biur Halacha brings a dispute as to whether one may ask a non-Jew to re-heat solid food that was previously cooked. Most poskim hold it is forbidden, but the Pri Megadim says that since in general one may ask a non-Jew to do a D’rabbanan for the sake of Shabbos, here one should not rebuke one who is lenient. He also brings that the Shaarei Teshuva in 318 ruled leniently. The Biur Halacha concludes that one may rely on this for the sake of Shabbos. As soon as the landlord saw that, he said, “This is a case of great need,” and called in a non-Jew to move the pot.
Afterwards, Rabbi Reisman said, “I don’t know why that was such a desperate situation – you could have shared my cholent.” “No,” the man replied, “it was a desperate situation because this is the third time in the last few weeks that I’ve made this mistake, and those times I was strict about it and we ate cold cholent. My wife was already mad at me, so imagine what she would have said this time!” Rabbi Reisman agreed that this was indeed a case of great need.