Gemara

(Rav Huna bar Chinena): R. Yosi is stringent about Tum'ah, for it has a basis in Torah. He is lenient about Techumim, for he holds that it is mid'Rabanan.

45b - Question: Why do we permit rain? Perhaps it came from clouds outside the Techum!

Answer: This is a Safek mid'Rabanan, therefore we are lenient.

Beitzah 2a (Mishnah - Beis Shamai): One may eat an egg laid on Yom Tov;

Beis Hillel forbid.

(Rav Nachman): The hen was designated for laying eggs. Beis Shamai hold like R. Shimon, who permits Muktzeh and Nolad. Beis Hillel hold like R. Yehudah, who forbids them.

Rav Nachman himself explained that regarding Shabbos, Beis Shamai hold like R. Yehudah, and Beis Hillel hold like R. Shimon.

3b (Beraisa): One may not move an egg laid on Shabbos or Yom Tov. A Safek is forbidden. If it became mixed with 1000, all are forbidden.

Question: Granted, Rabah forbids the egg due to Hachanah mid'Oraisa. Therefore a Safek is forbidden. However, Rav Yosef and R. Yitzchak forbid due to a decree. Every Safek mid'Rabanan is permitted!

Answer #1: The Seifa forbids a Safek Terefah egg.

Objection: Granted, if it discusses an egg laid on Yom Tov, we forbid a mixture even of 1000, for it is Davar she'Yesh Lo Matirim. However, a Terefah is Davar she'Ein Lo Matirim!

4a - Answer #2 (Rav Ashi): Really, it is a Safek if it was laid on Yom Tov or a weekday. It is a Davar she'Yesh Lo Matirim, so even mid'Rabanan it is not Batel.

24a (Beraisa - R. Gamliel): Safek Muchan is permitted;

R. Yehoshua forbids.

(Shmuel): The Halachah follows R. Yehoshua.

Shabbos 124b - Suggestion: Beis Shamai forbid Hotza'ah [on Yom Tov due to Shabbos], but perhaps they do not decree about Tiltul (moving Muktzeh)!

Rejection: Tiltul was forbidden due to Hotza'ah. If people can move, they will be Motzi!

151b (Mishnah): If a Nochri brought flutes on Shabbos, one may not eulogize with them unless they came from a nearby place.

Question: What is the meaning of '[they came from] a nearby place'?

Answer #1 (Rav): [We know that] it came from nearby (inside the Techum);

Answer #2 (Shmuel): We are Chaishinan (this will be explained) that it came from outside the wall.

Rishonim

Rif (Shabbos 65a): The Halachah follows Shmuel, who is concerned lest they were brought from outside the Techum. One must wait on Motzei Shabbos until they could come from a near place.

Rosh (Shabbos 23:9): Shmuel is lenient to assume that they came from outside the wall, and inside the Techum. Therefore, one may use them immediately on Motza'ei Shabbos.

Mordechai (Beitzah 685, brought in Beis Yosef OC 515 Kasav ha'Mordechai): Shmuel rules like R. Yehoshua. This implies that Shmuel forbids Muktzeh, and even Safek Muktzeh. Avi ha'Ezri says that even though we are stringent about Safek Muchan, this is because Muktzeh is [mid'Oraisa], like it says in Shabbos 124b, so we are not lenient. However, if it is not Safek Muktzeh, rather, a Safek Techumim, we are lenient. E.g. if a Nochri has a house inside the Techum and a house outside the Techum, and he brought a gift, it is permitted even for the one for whom he brought. We are lenient about Safek Techumim (Eruvin 45b), even though it is Davar she'Yesh Lo Matirim (it is totally permitted in another situation). This is unlike an egg [that we are unsure if it was laid on Yom Tov or on a weekday]. Rav Ashi forbids it due to Davar she'Yesh Lo Matirim. There, we leave it in its Chazakah, and say that it was not laid [until Yom Tov]. The same applies to a Safek if something was trapped on Yom Tov. This does not apply to a Safek about Techumim.

Tur (OC 301): Chachamim learned from "Im Tashiv mi'Sheves Raglecha..." about how to walk on Shabbos... and Tiltul (the Isur to move Muktzeh). Since one may not walk and talk on Shabbos like on a weekday, all the more so his Tiltul on Shabbos may not be like on a weekday, lest he come to move all his items from one corner to another.

Question (Bach 2): What is his source that Chachamim learned from Tiltul from this verse? In Shabbos 124b, we say that Tiltul was forbidden due to Hotza'ah. In Beitzah 37a, Rashi explains that Chachamim forbade Tiltul lest people come to Hotza'ah. They did not decree about Kelim and food, for people could not fulfill this, and they would not fulfill Oneg Shabbos and Simchas Yom Tov. The Mordechai says 'we forbid Safek Muchan because Muktzeh is mid'Oraisa, but we are lenient about Safek Techumim (Shabbos 34a, Eruvin 45b). In Eruvin 36a, we say that R. Yosi holds that Techumim has no source in Torah, so it is unlike Muktzeh, which is forbidden due to Hotza'ah. Avi ha'Ezri does not mean that Muktzeh is mid'Oraisa, rather, it is forbidden due to Hotza'ah, which is mid'Oraisa. Rabah holds that the Isur of an egg laid on Yom Tov is mid'Oraisa, due to Hachanah. Even the Amora'im who disagree, and hold that Chachamim decreed due to Peros detached [on Shabbos] or liquids that flowed from food [on Shabbos], forbid due to concern for a Torah Isur. This is unlike Techumim according to the opinion that Techumim is mid'Rabanan. Why does the Tur say that Chachamim learned from the Isur to walk and talk on Shabbos like on a weekday, which is based on the Navi's words "Im Tashiv mi'Sheves Raglecha..."? If so, also Muktzeh is not due to concern for a Torah Isur! Perhaps the Tur agrees that it is based on a Torah Isur, and in order to teach it with their laws [of walking and talking on Shabbos], he gave the reason he gave, or he found someone who said so. I later found that the Rambam (Hilchos Shabbos 24:12) explains the reason for Muktzeh just like the Tur says.

Tosfos (45b DH Iba'is): Even though this is Davar she'Yesh Lo Matirim, and Rav Ashi says that it is never Batel, even regarding a Safek mid'Rabanan [it is forbidden], Chachamim were lenient about Eruv.

Poskim

Shulchan Aruch (OC 325:5): If a Nochri trapped fish or detached Peros for himself, they are forbidden to a Yisrael. Even if it is a Safek whether he trapped or detached today, they are forbidden during the day. At night they are permitted immediately, even if he Vadai trapped or detached today.

Shulchan Aruch (7): If it is a Safek whether Peros were picked for a Yisrael, or if it is known that they were, but it is a Safek whether they were picked today, they are forbidden during the day and permitted at night bi'Chdei she'Ya'aseh.

Mishnah Berurah (32): If the two Sefekos came together, perhaps they were detached today, and perhaps they were for the Yisrael, it is forbidden on Shabbos, for even if we know that it was not for the Yisrael, Safek Muchan is forbidden on Shabbos. It is permitted immediately after Shabbos due to a Sefek-Sefeka.

Sha'ar ha'Tziyun (36): We are stringent on Shabbos only regarding Safek Muchan (Muktzeh), but not regarding other Sefekos. The Gemara explicitly says so.

Shulchan Aruch (ibid.): Some permit immediately at night.

Gra (DH v'Yesh): In Beitzah 4a we forbid a Safek mid'Rabanan because it is Davar she'Yesh Lo Matirim. Why are we lenient about rain? Tosfos (45b) says that we are lenient about Techumim. The Mordechai says that we are stringent about Davar she'Yesh Lo Matirim only if the Chazakah is that it was forbidden, e.g. Safek Nolad or Safek trapped, but not regarding Techumim. This requires investigation here. (Damesek Eliezer - the lenient opinion is unlike Tosfos, who forbids Davar she'Yesh Lo Matirim except for Eruvin, and unlike the Mordechai, for here the Chazakah is that it was not picked before Shabbos.) They must say that the Safek egg (3b) is forbidden only due to Safek Muchan (not due to Davar she'Yesh Lo Matirim), like it says on 24a. In any case we must distinguish from there.

Damesek Eliezer: All the Meforshim asked why Rav Ashi did not say that Safek Davar she'Yesh Lo Matirim is forbidden. Really Rav Ashi answered that the mixture is forbidden due to Davar she'Yesh Lo Matirim. The Safek is forbidden due to Safek Muchan. If it were due to Davar she'Yesh Lo Matirim, why does R. Gamliel permit Safek Muchan? Does he permit Davar she'Yesh Lo Matirim?! We never heard that Tana'im argue about this.

Shulchan Aruch (515:3): If a Nochri brought a gift for a Yisrael on Yom Tov, even if it is a Safek whether it was picked or trapped today, it is forbidden. If it is evident that it was not picked today, it is permitted for one for whom it was not brought. If it was from inside the Techum, it is permitted to all.