Gemara

(Beraisa - R. Meir): When the 14th falls on Shabbos, we destroy all Chametz before Shabbos.

13a (Beraisa - R. Elazar ben Yehudah Ish Bartusa): When Erev Pesach falls on Shabbos, we destroy everything before Shabbos, including Terumah. We leave enough Tahor Terumah for two meals that will be eaten before the fourth hour.

49a (Mishnah - R. Meir): When the 14th falls on Shabbos, we do Bi'ur before Shabbos;

Chachamim say, we do Bi'ur in its proper time.

R. Eliezer bar Tzadok says, we are Meva'er Terumah before Shabbos and Chulin in its time.

Rishonim

Rif and Rosh (5a and 1:14, b'Sof): When Erev Pesach falls on Shabbos, we destroy everything before Shabbos, including Terumah. We just leave enough Tahor Terumah for two meals that will be eaten before the fourth hour. R. Elazar Ish Bartusa says so. This is the Halachah.

Rif and Rosh (15b and 3:6, b'Sof): R. Elazar bar Tzadok says that when Erev Pesach is on Shabbos, we are Meva'er Terumah before Shabbos and Chulin in its time. The Halachah follows him, for he holds like R. Elazar Ish Bartusa.

Rambam (Hilchos Chametz 3:3): When Erev Pesach falls on Shabbos, we search for Chametz on the night before, i.e. Thursday night. One leaves enough Chametz to eat before the fourth hour on Shabbos. He destroys the rest before Shabbos. If Chametz remains on Shabbos after four hours, he is Mevatel it and puts a Kli over it until Motza'ei Yom Tov ha'Rishon, and does Bi'ur.

Ra'avad: The Rif did not say so. Rather, one does not destroy any Chulin until the time, like R. Elazar Ish Bartusa and R. Eliezer b'Rebbi Tzadok. Both of them hold like each other. This is primary.

Hagahos Maimoniyos (3): One who normally gives to a Nochri the bread that remains, should give it on condition that he not take it out [to Reshus ha'Rabim], like a scheme, and [only] a small amount. People with Yir'as Shamayim do so. One may not give to him on condition that he remove it.

Question: When Erev Pesach is on Shabbos, [after eating the morning meal,] may one tell a Nochri to take the tablecloth to his house? The Ri (Shabbos 19a DH Nosnim) permits giving food to a Nochri in the Chatzer, but not other objects, for surely he will take them out.

Answer (Teshuvas ha'Rosh 24:1): There is no need to give the tablecloth and bowls from the house. He shakes off the tablecloth, cleans the bowls with his finger, and hides them out of sight with other Kelim used for Chametz.

Poskim

Shulchan Aruch (OC 444:1): When Erev Pesach falls on Shabbos, we search for Chametz on the night before, i.e. Thursday night. We destroy everything before Shabbos, but leave enough Chametz for two meals on Shabbos.

Shulchan Aruch (4): After eating the morning meal, one shakes off the tablecloth they ate on, and cleans the bowls with his finger, and hides them out of sight with other Kelim used for Chametz.

Beis Yosef (DH Teshuvah): It seems that the Rosh discusses when there are small crumbs on the tablecloth. Therefore, he shakes it, and does not transgress for them, for they do not have a Shi'ur. Also, people step on them, and Bi'ur occurs automatically. One need not remove the tablecloth from his Reshus. For the same reason, he need not remove bowls from his Reshus. Rather, he cleans them well with his finger. What he cleans out is stepped on, and Bi'ur occurs automatically.

Magen Avraham (434:5): Crumbs are Batel automatically. Bitul is for [esteemed Chametz, e.g.] a nice bun.

Magen Avraham (442:10): If Chametz less than a k'Zayis is slightly dirty, one is exempt from Bi'ur.

Mishnah Berurah (15): One need not remove from the house small crumbs [that are shaken off the tablecloth] or what he cleans out of bowls. They will be trampled and automatically Batel. In any case it is good to sweep the house afterwards through a Nochri. If there is no Nochri, he sweeps it himself through a Shinuy (deviation, e.g.) with a garment.

Kaf ha'Chayim (32): All the more so one may sweep with a garment if the floor is of planks (i.e. and not earth) and there is no concern for evening out holes. After eating one must wash his mouth and clean his teeth, especially if there are holes in his teeth, for the Isur of Chametz during Pesach is any amount.

Shulchan Aruch (ibid.): If any bread remains, he may give it to a Nochri on condition that he not take it out to Reshus ha'Rabim.

Taz (5): Since we do not have Reshus ha'Rabim, one may give even much, but the Nochri may not take it outside the Techum.

Gra (DH li'Reshus): Siman 325:1 says like the Taz. This is like Shevus d'Shevus (telling a Nochri to do something forbidden mid'Rabanan), which is permitted for the sake of a Mitzvah (Eruvin 67b).

Magen Avraham (7): This means that he gives it to him Stam, and does not say to take it out to Reshus ha'Rabim. See 325:2.

Gra (DH Al): The Yerushalmi says like the Magen Avraham.

Kaf ha'Chayim (36): Chak Yakov and Mekor Chayim forbid Stam. Others permit.

Magen Avraham (7): Since we do not have Reshus ha'Rabim, one may command him to take it out. Maharshal says so. All permit in a city surrounded by a wall.

Mishnah Berurah (18): Some Poskim say that Reshus ha'Rabim is only where 600,000 people go every day. According to them, all our cities are Karmelis.

Mishnah Berurah (16): One may give an absolute gift to a Nochri. We permit this on Shabbos for the sake of a Mitzvah (306:3), i.e. so he will not have Chametz in his Reshus.

Kaf ha'Chayim (33): One may give only Chametz, but not other things, for it suffices to shake out the tablecloth and clean the bowls.

Kaf ha'Chayim (34): One may feed to animals or cast it into a privy. Also this is Bi'ur. We permit only to a dog, which he must feed, but not to a pig, which he need not feed.

Mishnah Berurah (17): One may not command him to take it out, for commanding a Nochri is Shevus. A Ba'al Nefesh (pious person) should be stringent and give it on condition that he not take it out. Hagahos Maimoniyos connotes like this.

Kaf ha'Chayim (35): One may not put into the Nochri's hand, for this is Akirah (uprooting from rest. This begins the Melachah of Hotza'ah.)

Shulchan Aruch (ibid.): He gives like a scheme, and a small amount.

Magen Avraham (8): This means that each time he gives to him a small amount. One may not give much at once, for then he clearly gives to him in order to take it out.

Gra (DH v'Davar): Shabbos 19a supports the Magen Avraham. It says that one may give food to a Nochri. Tosfos says that only food is permitted. The Rambam (Hilchos Shabbos 6:21) permits giving food to a Nochri on Shabbos to eat it.

Mishnah Berurah (19): Even though the Nochri intends to take it out, it is permitted because he does not explicitly command him.

Mishnah Berurah (20): A small amount is what he can eat in one meal. Any amount is permitted in Karmelis. If one has much Chametz in a special room and forgot to sell it before Shabbos, one must give it for a gift to a Nochri. One may give it to a Nochri he knows, who will surely return it to him after Pesach. He must be Makneh (transfer ownership) to him through a Kinyan that works, i.e. Hagbahah (lifting it) or Meshichah, i.e. the Nochri takes it to his Reshus. If there is too much Chametz to be Moshech, he is Makneh to him through handing over the key. If he fears to give a gift, lest the Nochri not return it, some Acharonim permit selling it to him. This is not called buying and selling on Shabbos, since he does do only to save himself from the Isur of Chametz. Several Acharonim disagree. They forbid a sale in every case, even if he does not take money from him at all, just they fix a price, or he sells for the market price without specifying the price. Even though he sells for the sake of Mitzvas Bi'ur Chametz, Chachamim did not permit an absolute sale even for the sake of a Mitzvah (306:6).

Kaf ha'Chayim (38): Even if a sale is permitted, the Nochri should not write a document. He should put the money in a box, and not in the Yisrael's hand. Eshel Avraham permits a document in the Nochri alphabet. Simlas Binyamin says to rely on this only to avoid a big loss. R. Zalman forbids even to avoid a big loss. He permits only a gift.

Kaf ha'Chayim (39): One may give a small amount even 100 times (if the Nochri keeps taking it out and returning).

Shulchan Aruch (5): If Chametz remains after eating [in the morning], he is Mevatel it and puts a Kli over it until Motza'ei Yom Tov ha'Rishon, and does Bi'ur.