Gemara

Mishnah - Beis Shamai: One may not Motzi a child, Lulav or Sefer Torah to Reshus ha'Rabim (on Yom Tov);

Beis Hillel permit.

A reciter of Beraisos: If one was Shochet an Olas Nedavah on Yom Tov, he is not lashed.

R. Yitzchak bar Avodimi: This is like Beis Shamai, but Beis Hillel say that Mitoch (since) Hotza'ah is permitted for a need (of Yom Tov), it is also permitted without a need, and similarly regarding Shechitah!

Question (Rabah): Perhaps Beis Shamai and Beis Hillel do not argue about Mitoch, rather, about whether or not the Isur Hotza'ah and the need to make Eruvin applies only to Shabbos, or even to Yom Tov;

Beis Hillel learn from "V'Lo Sotzi'u Masa mi'Bateichem b'Yom ha'Shabbos" - the Isur is only on Shabbos, not on Yom Tov.

Answer (Rav Yosef): If so, they would have argued about rocks!

The following shows that also R. Yochanan holds that they argue about Mitoch.

A reciter of Beraisos: If one cooked Gid ha'Nasheh in milk on Yom Tov and ate it, he receives five sets of lashes. He is lashed for cooking a Gid (that may not be eaten) on Yom Tov, for eating a Gid, for cooking meat and milk, for eating meat and milk, and for burning on Yom Tov.

R. Yochanan: Do not teach this in the Beis Medrash! No one is Mechayev for burning or cooking on Yom Tov;

If someone is Mechayev, it is Beis Shamai, who do not say Mitoch regarding Hotza'ah, and likewise regarding burning (and cooking);

Beis Hillel say Mitoch regarding Hotza'ah, and likewise regarding burning (and cooking).

16b - Beraisa: One may make Eruv Chatzeros on Yom Tov for Shabbos, because we do not forbid someone to prepare for Shabbos with something that is permitted today.

Rishonim

Rashi (DH Liflegu): Muktzah was forbidden lest people transgress Hotza'ah. Rav Yosef says that if Beis Hillel totally permitted Hotza'ah, the Mishnah should have said that Beis Hillel permit even to move rocks!

Tosfos (DH Hachi): Rav Yosef says that they did not argue about Hotza'ah of rocks, for all Mechayev. They argue about Hotza'ah which is not necessary for eating, but is a need of the day.

Rif and Rosh (1:18): Beis Shamai do not say Mitoch that Hotza'ah is permitted for a need, it is also permitted without a need; Beis Hillel say Mitoch. 'For a need' means for a need of food; without a need refers to a child, Lulav or Sefer Torah. The same applies to Kelim. However, even Beis Hillel forbid rocks and similar things. This is the Halachah.

Ran (DH Gemara): The Acharonim permit Hotza'ah only if there is some need for the day. Rashi permits as long as it is not for the sake of after Yom Tov. The Rif leans to agree.

Rif and Rosh (ibid.): We hold that there is no need for Eruv Chatzeros or Shituf Mavo'os on Yom Tov, but they are needed for Yom Kipur.

Ran (DH v'Kaima): The Rif connotes that there is no need at all for Eruv Chatzeros on Yom Tov. However, the Rashba forbids to take Kelim from Chatzer to Chatzer without an Eruv. Just like there is an Isur mid'Oraisa to take them from Reshus ha'Yachid to Reshus ha'Rabim, it is Asur mid'Rabanan to take them from one Chatzer to another. The Gemara (16b) permits to make Eruv Chatzeros on Yom Tov, for this matter is permitted today. This sounds like there is no need at all for Eruv Chatzeros on Yom Tov. Really, it refers to Kelim that are needed for Yom Tov. Perhaps since most Hotza'os do not require an Eruv, Chachamim did not decree at all. The Rif connotes like this. However, the Gemara favors the Rashba.

Rambam (Hilchos Yom Tov 1:4): If on Yom Tov one did any Melachah for which one is liable on Shabbos, if it was not for the sake of eating he is lashed. The only exceptions are Hotza'ah and burning. Therefore, one may take a child, Sefer Torah, key or similar things from Reshus to Reshus. Similarly, one may burn even if not needed for eating.

Magid Mishnah: Rashi says that carrying rocks is forbidden mid'Rabanan, for it is unnecessary exertion. It seems that the Rif and Rambam agree. The Rambam did not need to specify here, for he says (Halachah 17) that whatever may not be moved on Shabbos is Asur on Yom Tov, unless it is for eating. The Rambam (Eruvin 8:4) holds that there is no need for Eruv Chatzeros on Yom Tov, like the Rif.

Rambam (5): Chachamim forbid to do on Yom Tov any Melachah that could be done before Yom Tov without a loss, even if it is for eating. This is a decree lest people leave Melachos that could be done before for Yom Tov, and he will refrain from Simchas Yom Tov.

Rambam (6): Chachamim did not forbid Hotza'ah on Yom Tov, even though it could be done beforehand, because they wanted to increase Simchas Yom Tov. A person may take with himself whatever he needs.

Rosh (1:18): R. Chananel says that the Mishnah permits taking a child to circumcise him, a Sefer Torah to read and a Lulav to fulfill the Mitzvah. Mitoch permits only things that are somewhat needed, e.g. for a Mitzvah. If we would not say so, one who cooked on Yom Tov for the sake of Chol would not be lashed.

R. Tam permits to take a child for other needs, e.g. if one wants to go to the Beis ha'Keneses or for a walk for Simchas Yom Tov and he cannot leave the child alone at home. One may take Seforim to the Beis ha'Keneses and return them (even if he will not use them again that day). We permit returning them, so people will take them to begin with.

The Ge'onim say about any Hotza'ah permitted for eating, it is similarly permitted for decoration, for Kavod Yom Tov. But to carry for neither of these is forbidden. It is improper to go out with keys (not for food) hanging from one's belt.

Poskim

Shulchan Aruch (518:1): Mitoch that Hotza'ah is permitted (on Yom Tov) for needs of eating, it is also permitted without a need, e.g. a child, Sefer Torah, Lulav or Kelim.

Rema: This is if he needs them somewhat, or if he fears lest they be stolen or another loss.

Beis Yosef (DH u'Mah she'Chosav Rabeinu): The Tur forbids a key if there is no need for it today. However, SMaK permits a key to a money chest. This is considered somewhat of a need, since (if he may not take it) he will worry lest he suffer a loss. Similarly, R. Peretz permits to return a Machzor from the Beis ha'Keneses, even though it is not needed until Motza'ei Yom Tov. Since we live among Nochrim, he would worry otherwise. Ula did not list this among things in which the end is permitted for the sake of the beginning, because it is not explicit in the Mishnah. Rabeinu Me'ir forbids if he will not use them again that day.

Kaf ha'Chayim (6): Some say that the Shulchan Aruch permits without any need at all. Others say that the Shulchan Aruch does not discuss this, for the Beis Yosef was undecided. The Rema explains that the Shulchan Aruch's intent is to forbid.

Taz (1): I do not know why the Rema rules like SMaK, for most Rishonim and Acharonim disagree. The Mordechai says that if one brought a Machzor to the Beis ha'Keneses he may return it, but if it was there from before Yom Tov it is forbidden. We accept the Rema's ruling, but a Ba'al Nefesh should be stringent.

Beis Yosef (DH ul'Inyan): The Magid Mishnah says that the Rif and Rambam hold like Rashi, that carrying rocks is forbidden mid'Rabanan, for. Also the Ran says that the Rif holds like Rashi; presumably, the Rambam holds like his Rebbi (the Rif), therefore we follow this. Anything totally unnecessary is forbidden mid'Rabanan, even though the Ran connotes that Rashi permits even mid'Rabanan.

Mishnah Berurah (6): Many Poskim allow Hotza'ah only for eating, Mitzvos or needs of the day. One should be stringent. If one can leave the key to a money chest with a trustworthy person who is staying at home, all forbid to carry it.

Beis Yosef (DH Kosav ha'Ran): An Eruv permits to take things from Chatzer to Chatzer.

Mishnah Berurah (10): The Yam Shel Shlomo says that we remove knives (which people used to carry in their garments) before leaving the city (for one will not need it), but one need not remove them before leaving the Eruv within the city. Some permit even outside the city, for perhaps one will want to cut a fruit. However, if surely it is not needed, e.g. when going to the Beis ha'Keneses, it is forbidden.

Mishnah Berurah (3): Even if the child could stay home with his mother, if one enjoys his company it is Simchas Yom Tov to take him.

Shulchan Aruch (ibid.): Rocks and similar things are forbidden.

Rema: One may play with a ball even in Reshus ha'Rabim, even though it is mere recreation. If one made an Eruv, he may move and be Motzi any Keli, even if it is not needed for the day at all.

Taz (2): The Maharshal was astounded that the Rema permits balls even to adults. This is children's folly, not Simchas Yom Tov! If he could Mevatel this, he would. One with Yir'as Shomayim should make Eruv Chatzeros for Yom Tov, just like for Shabbos; one should not change the custom (if people do not). Since our custom is to make an Eruv on Erev Pesach for the entire year, one should say that it is also for Yomim Tovim. If one can fulfill all opinions without a loss, he should.

Kaf ha'Chayim (14): One who makes an Eruv on Erev Shabbos should say that it is also for Yom Tov (if Yom Tov is coming that week). Some permit to carry wherever an Eruv would permit, even if an Eruv was not placed.

Mishnah Berurah (10): It is not clear why the Rema did not comment in Siman 528, where the Shulchan Aruch says that there is no need at all for Eruv Chatzeros.

Shulchan Aruch (528:1): There is no need for Eruv Chatzeros on Yom Tov.