Can one ask a non-Jew to turn off a propane tank on Shabbat? Effectively turning off the fire. Are you really benefiting from it other than safety?
There are two issues with amirah legoy, one is directing a gentile to do a melakhah for you regardless of any benefit, since you are causing the melakhah to be done. The other is a problem of benefit, which is a problem even if he does it on his own, as long as he is doing it for you (Chazal forbade benefit lest you come to ask him next time).
So, the problem isn’t benefit, it is amirah. Turning off propane is a derabanan, not real kibuy, since there is no coal being made – and possibly also only a grama, since unlike your stove, I don’t think the flame goes out immediately, because there is still propane left in the tube which is being fed to the flame (correct me if I am wrong). Anyway, it is certainly a shevut dishvut (amirah on a derabanan) and maybe even less than that, which is absolutely permissible for safety issues.
Is one allowed to ask a non-Jew to turn it on? My shul is most likely going to daven outside all winter because of COVID. It seems that it fits the two circumstances (illness and mitzvah needs) under which amira le’akum is permitted. Do we distinguish with de’orayta/derabanan here? In terms of illness, it seems like we can apply the idea that “all people are considered ill in cold conditions” and with mitzvah needs you would be enabling davening betzibur, talmud Torah, etc. What am I missing here — it still feels that it should be assur. If mutar, how should one go about asking/paying the non-Jew?
As you note – הכל חולים אצל הצינה – which means that one has the status of a choleh kol gufo, and amirah is permissible even for a de’orayta. It’s an explicit Shulchan Arukh, if it really is cold – Shulchan Arukh OC 276:5,
בארצות קרות, מותר לא”י לעשות מדורה בשביל הקטנים ומותרין הגדולים להתחמם בו, ואפי’ בשביל הגדולים מותר אם הקור גדול, שהכל חולים אצל הקור; ולא כאותם שנוהגים היתר אף על פי שאין הקור גדול ביום ההוא.
See Mishnah Berurah (276:41) who states that the difference between adults and children is whether it is a real difficulty for them to be this cold. So once it is, it is permitted for adults as well.
The principle of amirah legoy for a de’orayta for choleh kol haguf, is in Shulchan Arukh OC 328:17,
חולה שנפל מחמת חליו למשכב ואין בו סכנה, הגה: או שיש לו מיחוש שמצטער וחלה ממנו כל גופו שאז אף על פי שהולך כנפל למשכב דמי (המגיד פ”ב), אומרים לא”י לעשות לו רפואה אבל אין מחללין עליו את השבת באיסור דאורייתא
See Mishnah Berurah 328:47,
לעשות לו רפואה – אפילו במלאכה דאורייתא וה”ה שאר צרכיו כגון לאפות ולבשל וכיוצא באלו אם צריך לכך:
All of this is WITHOUT the need to refer to mitzvah, rabim, etc. Of course all of that makes it better but it is not necessary in this case.
As to your question of payment, etc., – if you can ask him to do it for the sake of a choleh, you can ask him on Shabbat. There is no issur of sekhar Shabbat if the person receiving it is a non-Jew (just don’t pay him on Shabbat!).