Just as on Shabbat the Sages prohibited a Jew from asking a non-Jew to do melakha for him, so too on Ḥol Ha-mo’ed they prohibited a Jew from asking a non-Jew to do melakha for him that he would not be allowed to do himself (SA 543:1; MB ad loc. 1). This is the case even if the Jew anticipates that if he does not hire the non-Jew to do the work on Ḥol Ha-mo’ed, he will have to pay him more after Ḥol Ha-mo’ed, for the permissibility of preventing a loss applies only when one would lose something that he already has (Ḥayei Adam, Yom Tov, 106:12; MB ad loc. 2).
One may ask a non-Jew to do melakha that a Jew may do – i.e., labor that is unskilled or undertaken with a shinui. An example would be sewing clothing for the festival (sections 7-8 above). If the non-Jew wants to do the work in his usual professional manner, he does not have to be asked not to do so (see SSK ch. 68 n. 137). Similarly, if a non-Jew decides on his own to do melakha for a Jew on Ḥol Ha-mo’ed, the Jew may benefit from it (Piskei Tosafot; Kaf Ha-ḥayim 543:5).
Even though a Jew may not ask a non-Jew to do work for him which is prohibited on Ḥol Ha-mo’ed, it is permissible if it will lead to a mitzva being performed on Ḥol Ha-mo’ed. Therefore, a non-Jew may be asked to finish building a synagogue so Jews may pray there (MB 543:1; SHT 544:10).
One may give a non-Jew work before the festival, on two conditions: 1) The non-Jew is working for himself – for example, he works as a contractor who is paid by the job (not by the hour), and he decides when he will do the work. Alternatively, the non-Jew shares in the profits of the work, so he is working for his own benefit. 2) It must not be work that is normally paid by the hour, so that no one will suspect the Jew of hiring the non-Jew to work on the festival. Since buildings today are normally built by contractors, it is permissible for a non-Jew to continue building a house for a Jew on Ḥol Ha-mo’ed (SA 543:2; BHL 244:1 s.v. “o liktzor”).
If there is merchandise that a Jew may not buy on the festival, he may not ask a non-Jew to purchase it on his behalf either, even if he makes the request before the festival (BHL 539:1 s.v. “bein liknot”). However, one may tell a non-Jew, “You should buy it for yourself. I will buy it from you later, and you will make a profit.” One may even lend the non-Jew money to do so (SA 307:3; MB ad loc. 13; SSK 68:34).
On Ḥol Ha-mo’ed, a Jew may give a non-Jew work to do if he makes it conditional upon the work being done after Ḥol Ha-mo’ed. The Jew may not weigh, measure, or count the items which he is leaving with the non-Jew, because that is a weekday activity. If the non-Jew later ignores the terms and does the work on Ḥol Ha-mo’ed, the Jew does not have to object, since he explicitly requested that he do the work after the festival (SA and Rema 543:3).
Some say that a Jew may not cook or do other melakhot on Ḥol Ha-mo’ed for a non-Jew, as all the melakhot permitted on Ḥol Ha-mo’ed are permitted for the sake of enjoying the festival. Since a non-Jew has no mitzva to enjoy the festival, a Jew may not do any melakha for him (Ḥayei Adam 106:11). Others allow it (see Shevet Ha-Levi 8:124:2). In practice, when there is a great need, such as to sanctify God’s name or prevent animosity, those melakhot may be done for a non-Jew on Ḥol Ha-mo’ed (see SSK 68:37 and Harḥavot here).