The prohibitions relating to boundaries (teḥumin) apply on Yom Tov just as they do on Shabbat. The idea behind these holy days is to let the Jews rest from their travails and worries and enable them to engage in Torah study and mitzva-oriented simḥa. Therefore, the Sages ordained that a person should not go outside his teḥum, which is defined as wherever he is at the start of Shabbat or Yom Tov (mekom shevita), plus 2,000 amot in each direction (approximately 0.57 mi or 912 m). If one is spending Shabbat or Yom Tov in a field (i.e., not in a city or town), his mekom shevita is four amot plus 2,000 amot in each direction. If he is spending Shabbat or Yom Tov in a city or town, the entire settled area (or area enclosed by an eruv) is considered his mekom shevita. He may travel 2,000 amot in each direction beyond that area. These laws are explained in detail in Peninei Halakha: Shabbat, vol. 2, ch. 30.
Even though on Yom Tov it is permitted to do melakha necessary for food preparation, one may not go beyond the teḥum even for this. The permission to engage in food preparation on Yom Tov is limited to food already in one’s possession. However, just as one may not harvest grain or hunt animals on Yom Tov, so too he may not go outside the teḥum to get food (Ramban, Milḥamot Hashem, Beitza 23b; Rashba, Avodat Ha-kodesh 1:1). Furthermore, the prohibition of teḥumin is not one of the 39 categories of forbidden melakha, and therefore it is not included in the permission to engage in melakha for food preparation (Ḥatam Sofer, OḤ §149).
If a non-Jew brought fruit from outside the teḥum, then according to Torah law, as long as he brought the fruit for himself or for another non-Jew, any Jew may eat them. However, the Sages declared that teḥumin apply even to objects belonging to non-Jews (SA 401:1). Since the fruits that the non-Jew brought have left their teḥum, a Jew may not carry them more than four amot. If the non-Jew brought them into a home or a site that is enclosed by a fence or an eruv, a Jew may carry them within the enclosed area.
If the non-Jew brought the fruit for a Jew, the Sages forbade the intended recipient and the members of his household from eating them until enough time has passed after Shabbat for the fruit to have been brought then. However, other Jews may enjoy the fruit even on Shabbat, as long as they do not carry the fruit more than four amot (SA 325:8).
If a Jew brought food from outside the teḥum knowing full well that doing so was forbidden, no Jew may enjoy the food on Yom Tov.
. Just as one may not go beyond the teḥum for food, so too one may not go beyond the teḥum for mitzva items such as a lulav or shofar. However, one may ask a non-Jew to bring him such an object to enable him to fulfill the mitzva. As we have already seen, the Sages permitted transgressing a shvut di-shvut (double rabbinic prohibition) for the sake of a mitzva or for a great need. Since, according to most Rishonim, teḥumin is rabbinic, asking a non-Jew to bring the object renders it a shvut di-shvut, and thus permitted for the sake of a mitzva. Although the Sages declared that it is forbidden to benefit from something that a non-Jew brought from outside the teḥum, which would seem to render such an object forbidden, there is a principle that “mitzvot were not given for benefit,” so the Sages’ enactment does not apply to mitzvot (SA 555:1: MB ad loc. 3; Peninei Halakha: Shabbat 9:11).
. See Peninei Halakha: Shabbat 26:2, where we explain that if a Jew knowingly performs a melakha, then even though the prohibition is rabbinic, it is forbidden for any Jew to benefit from it. If he brought the food unknowingly in a way that violated Torah law (such as by driving), according to most poskim it is forbidden to benefit from it (SA 318:1). Others maintain that in pressing circumstances one may benefit (MB 318:7). If he brought the food unknowingly in such a way that there was no violation of a Torah law, such as if he came by foot from outside the teḥum, it is permitted to benefit from it (BHL 318:1 s.v. “ha-mevashel”; Peninei Halakha: Shabbat 26:3).