Detailing the final three exceptions when one can use a non-Jew's help on Shabbos and Yom Tov, and how these can be used.
BEIN HASHMASHOS, I.E. DURING TWILIGHT
Under certain circumstances, it is permitted to ask a non-Jew to perform even a biblically forbidden action during Bein Hashmashos. This is the period from the start of Shabbos or Yom Tov at sunset until nightfall (Shulchan Aruch O.C. 261:1). In this context the exact end length of time varies by location and is approximately 30 minutes after sunset (see Igros Moshe O.C. 4:62).
Although all Shabbos and Yom Tov prohibitions take effect before sunset, our Sages allowed Amira L'Akum, asking a non-Jew to help, during the 'twilight' period for very necessary needs. This includes anything that if not done will cause stress or anguish on Shabbos (Mishna Berura 261:16). Examples include that one may explicitly ask a non-Jew to turn on a light or an appliance or to wash clothing that’s needed for Shabbos (no hint needed). Also, in case one did not light Shabbos candles on time, one may ask a non-Jew to light the candles during this Bein Hashmashos period.
P’SIK REISHA, I.E. WHEN THE MELACHA IS NOT DESIRED
P’sik Reisha refers to an ancillary consequence that will invariably happen as a result of an action. If an action itself is essentially permissible, but will inevitably result in a Melachah being performed, it is permitted to ask a non-Jew to perform the action (Rema O.C. 337:2, Magen Avraham O.C. 276:10, Mishna Berura 253:51).
For example, if one needs to access a refrigerator on Shabbos, but opening the refrigerator door will cause the light inside to turn on, one may not open the door. However, one may explicitly ask a non-Jew to open the door since the desired action of retrieving food is permitted. Alternatively, if one left an item that is necessary for Shabbos in a car, one may ask a non-Jew to retrieve it, even if the car light will turn on. This is because the act of retrieving the item itself is permitted and it's only the means that is a forbidden act.
In a similar vein, one may ask a non-Jew to do a permissible activity, even if it is clear that the non-Jew will do it in a prohibited way. For example, one may ask a non-Jewish cleaner to wash dishes even if they are likely to use hot water. This is because the action of washing up itself is a permitted activity and the non-Jew is deciding to do it in a forbidden way for their own convenience (Shemiras Shabbos Kehilchasa 30:24). In contrast, one may not ask a non-Jewish helper to mop on Shabbos since the act of mopping itself is inherently forbidden.
Where the act is inherently permitted but it is not reasonable to expect the non-Jew to do it in a permitted way, the Poskim debate if this is allowed or not. For example, asking a non-Jew to bring food to the tenth floor which means that they will obviously use the elevator and not the stairs, some allow it (Shemiras Shabbos Kehilchasa 30:25) but some forbid it (Rav Elyashiv quoted in Shvus Yitzchok v.17 p.52, Orchos Shabbos 23:53).
As long as we don't ask, we may benefit from a Melachah performed by a non-Jew on Shabbos if the benefit is indirect, i.e. brought about by the removal of something (Shu"A HaRav 305, Chayei Adam 62:3, see Piskei Teshuvos 276:2). For example, shutting off a light or opening an envelope are considered indirect benefits via the removal of impediments, which means we may benefit from a non-Jew doing this, even if done on our behalf.
It's important to note that whilst we may not ask them to do such an action, one may hint to the non-Jew indicating the desire for the action to be done. e.g. 'We need to sleep, but can't because the light is on'. The hint must be a passive statement and may not contain any mention of an action, so saying something like 'Can you help' or 'Come to my house' would be forbidden (Peri Megadim Eshel Avrohom 307:27).
One should only use this for the sake of a Shabbos or a Mitzvah need. In addition, it is forbidden to ask or hint to a non-Jew to do anything that we could have done before Shabbos or Yom Tov (Piskei Teshuvos 276:2).
We will BeH conclude this topic and explain the reasons for the prohibition of Amira L'Akum next week.