Gemara

6a (Rav Yehudah): If one finds Chametz in his house on Yom Tov, he covers it with a Kli.

(Rava): If it is Hekdesh, he need not cover it.

This is because [the entire year] people stay away from Hekdesh.

(Rav Yehudah): One must make a wall 10 Tefachim tall in front of Chametz of a Nochri [in his house. One does not need a wall for Chametz of Hekdesh [because people stay away from Hekdesh].

10b (Mishnah -Chachamim): If one did not check on the night of the 14th, or the next morning, or during the festival, he checks after the festival.

(Beraisa - R. Yehudah): If one did not check at one of these three times, he does not check later.

R. Yehudah decrees not to check after it is forbidden, lest one find Chametz and eat it. Chachamim do not decree.

(Abaye): R. Yehudah holds that people do not stay away from Chametz. (It is permitted the rest of the year.)

(Rava): Chachamim do not decree about Chametz. Since he searches in order to burn it, we are not concerned lest he eat [what he finds]!

Chulin 103b (Mishnah): One may not bring cheese on the same table with any kind of meat.

104a - Inference (Rav Yosef): The Torah forbids fowl cooked with milk. If not, we would not make a decree (forbid putting them on the same table) to guard a decree (eating them)!

Rebuttal (Abaye): If we allow fowl and milk on the same table, one may come to put beef and milk on the same table, and eat them together!

(Mishnah): A Zar (non-Kohen) may eat at the same table with a Kohen eating Chalah of Chutz la'Aretz.

(Rav Yosef): This shows that we do not decree due to a decree!

Question (Abaye): That would be a proof if the Mishnah discussed Chalah of Chutz la'Aretz in Eretz Yisrael, where there is reason to decree, lest a Zar or a Tamei ignoramus Kohen eat mid'Oraisa Chalah [of Eretz Yisrael];

However, in Chutz la'Aretz, there is no concern for any Torah Isur [of Chalah]. There is no need to decree!

107b (Mishnah - R. Shimon ben Gamliel): Two lodgers may eat at the same table, one meat and the other cheese, and we are not concerned.

Nedarim 41b (Mishnah): One may eat with one from whom he is Mudar Hana'ah (may not benefit from him, due to a vow) at a table.

Rishonim

Perush ha'Rosh (41b DH v'Ochel): We permit eating with one from whom he is Mudar Hana'ah. Since he vowed from him, presumably they hate each other more than two guests. A guest may eat meat on the same table with another guest eating cheese.

Ran (Chulin 32b, top): Perhaps we are more stringent about meat and milk than other Isurim because people do not stay away from it, for each is permitted by itself. One may not bring cheese with meat on a table, unlike other Isurim. Therefore, if bread was baked in an oven with roasting meat, one may not eat it with a dairy dip, since it has the smell of meat. This is a mere stringency. Even Levi and Rava, who hold that smell is not significant, forbid.

Rashba (1:177): One need not make a wall in front of a Nochri's Chametz, unless he deposited it with him. Then, we are concerned lest he forget and eat it, since it is handed over to him and he does not stay away from it. When the Nochri holds his Chametz, he guards it and the Yisrael stays away from it. This is like Chametz of Hekdesh. Due to the severity of Hekdesh, people are careful and stay away from it. Surely one may not put it on the table with him. He would be likely to transgress, since they know each other. Just like the Yisrael would not mind if the Nochri eats from his (the Yisrael's) food, the Nochri would not mind if the Yisrael eats from his food. We forbid even if they eat on separate cloths, since Chametz forbids [a mixture] b'Mashehu (any amount). It is difficult to avoid a crumb of the Nochri's Chametz from entering the Yisrael's food. We tell a Nazir to stay far from a vineyard! (Everyone must distance from transgression.)

Tosfos (Pesachim 2a DH Ohr): The Ri says that even though Bitul Chametz suffices, Chachamim were stringent to require Bedikah lestone comes to eat it, for it is permitted the entire year. People do not stay away from it, unlike other Isurei Hana'ah.

Poskim

Shulchan Aruch (YD 88:1): One may not put even meat of Chayos or fowl on a table on which one eats cheese, lest he come to eat them together.

Taz (1): The Gemara says that in a case like this, we decree a Gezeirah li'Gezeirah.

Kaf ha'Chayim (6): Since it is common to eat them together, it is like one Gezeirah (Levush).

Shach (1): Obviously, similarly one may not bring cheese on a table on which he eats meat.

Shach (2): We forbid only meat on a table on which one eats cheese, or vice-versa, for people do not distance from them, since each is permitted by itself. One may put Nevelah on a table with Kosher meat. The Ran clearly says so. However, the Rosh connotes that the same applies to other Isurim. Perhaps since [something Asur due to] a Neder is forbidden only due to his vow, it is like meat and milk, in which each is permitted by itself. The Rosh could agree about other Isurim. The Shulchan Aruch (OC 440:3) rules like the Rashba, that if a Nochri brings Chametz into a Yisrael's house, he may not put it on the table with him, even on a separate cloth. This is due to the stringency of Chametz, which is b'Mashehu. A. crumb of Chametz is very likely to enter the Yisrael's food. However, the Beis Yosef (440) brought Orchos Chayim, who connotes unlike this. He permits the Nochri to eat Chametz in the Yisrael's house, perhaps even on the table on which the Yisrael eats Matzah. He permits because 'this is like' two guests. This connotes that he [normally] forbids even when each is forbidden by itself. I agree with the Ran. All the Poskim connotes like him. They brought the Isur [to bring on a table together] only for meat and milk. I say that Orchos Chayim agrees. There is different, for man lives through eating bread. He is used to it the entire year, and does not stay away from it. The Rosh (Avodah Zarah 2:15) writes so about Bishul Akum (food cooked by Nochrim). Bread requires more distancing, for it is the staple food. The Beis Yosef brought this (Reish YD 114). The Ran himself agrees about bread. If Chalah of Chutz la'Aretz were mid'Oraisa, we would have decreed not to put it on a Zar's table! (See 323:2.) Even though it is forbidden by itself, there is more reason to decree about bread. Based on this, one may not bring Asur bread on the table, just like we forbid meat and milk.

Sifsei Da'as: The Shach forbids bringing on a table something that people do not stay away from. The same applies if one is Mudar Hana'ah from something, or forbids himself to eat it. If he stays away from it, it is permitted. Minchas Yakov (77:21) says that people do not stay away from Chametz (Pesachim 11a). Perhaps they do not refrain also from Asur bread. See [Tosfos at] the beginning of Pesachim.

Kaf ha'Chayim (5): The Birkei Yosef and Aruch ha'Shulchan are no more stringent about bread, but one should be stringent like the Sifsei Da'as.

Pischei Teshuvah (1): Be'er Heitev of the Maharit says that that Or Zaru'a forbids lighting a Ner of butter, lest it drip on meat. Beis Lechem Yehudah says that if so, we forbid also lard [lest it drip on cheese]. The Pri Megadim distinguishes. Melted butter is prone to spill out of a Ner. We are not concerned lest lard drip from a Ner made like a candle. Siman 433:2 (regarding Bedikas Chametz) connotes unlike this. I distinguish. There, he must enter it in holes and cracks, so we are concerned even for a Ner like a candle. They did not mention not to light with Chelev, for people stay away from it and are careful not to hold it near food, unlike butter.

Kaf ha'Chayim (7): The Birkei Yosef says that where people use candles of Chelev, one should not bring it on a table or where they prepare food.

Gan ha'Melech (71, cited in R. Akiva Eiger): Chachamim forbade lighting Ner Shabbos with certain wicks and oils [that do not burn well, lest one come to tilt the Ner to make it burn better.] There is no Heter through a Shomer (someone to guard that he not tilt the Ner). It needs constant guarding. One cannot avoid a lapse of attention for a moment! This is why we forbid putting meat and cheese on the table together. A Shomer does not help, since it must be constant without any lapse, and this is impossible.

Kaf ha'Chayim (16): A Shinuy (deviation) helps [to remind him not to mix them] only if there are two people. Therefore, a Shomer helps with a Shinuy.

Shulchan Aruch (ibid.): One may put them next to each other on a table on which one arranges the courses.

Kaf ha'Chayim (8): They must be distanced so they will not splash onto each other.