Gemara

(Mishnah): One may eat a Shelishi (third degree Tum'ah) with Terumah mixed in.

Question (Rav Hamnuna): If one who eats a Shelishi becomes a Shelishi, why is this permitted?

Answer: Less than a k'Zayis of Terumah is mixed with the volume of half a loaf.

98a: A half-k'Zayis of Chelev fell into a pan of meat. Mar bar Rav Ashi thought that if there was 30 times as much Heter, it would be Mevatel the Chelev.

Rav Ashi: Do not be lenient about the quantities that Chachamim fixed (i.e. 60 times);

Also, you must be stringent because R. Yochanan forbids mid'Oraisa even Chetzi Shi'ur (less than the quantity for which one gets lashes or Kares, i.e. k'Zayis).

105a (Rav Chisda): If one ate meat, he may not eat cheese (right away)

Question (Rav Acha bar Yosef): Is meat stuck between teeth considered meat (to forbid one to eat cheese)?

Answer (Rav Chisda): "The meat was still between their teeth." (It is considered meat.)

(Mar Ukva): I do not compare to the piety of my father. He would not eat cheese for 24 hours after eating meat, but I eat cheese at the next meal!

Gitin 64b (Rav Chisda): A minor cannot acquire for others.

Question (Rav Chinena Vardan - Mishnah): An Amah ha'Ivriyah (who is a minor) can acquire wine on behalf of for others for the sake of Shituf Mavo'os (a kind of Eruv).

Answer: Shituf Mavo'os is mid'Rabanan. Chachamim were lenient to allow her to acquire, but mid'Oraisa, she cannot acquire.

(Rav Chisda): Rav Chinena was silent. (He should have said that Chachamim enacted like Torah law!)

Rav Chinena did not say so, for he holds that only enactments with an origin in Torah (e.g. to take the Lulav every day of Sukos) are patterned on Torah laws.

Rishonim

Rambam (Hilchos Berachos 1:19): Anyone who eats something forbidden, whether b'Mezid or b'Shogeg, does not bless before or after on it. This applies to Tevel mid'Rabanan... and all the more so Neveilah, Tereifah...

Rebuttal (Ra'avad): The Gemara said that we do not make a Zimun with one who ate Isur, for this is not considered Keva (fixed). Why shouldn't he bless? He benefited!

Rosh (Berachos 7:2): Sanhedrin 6b proves that the Ra'avad is correct. If one stole wheat, ground it, baked it, separated Chalah, his Berachah is really blasphemy. This implies that he must bless, just it is blasphemy.

Tosfos (Berachos 45a DH Achal): Eating Tevel is not considered eating. If one stole wheat, ground it, baked it and blessed on it, his Berachah is really blasphemy.

Rambam (Hilchos Ma'achalos Asuros 9:28): If one ate meat, he may not eat milk afterwards until waiting the (normal) time in between until another meal, which is about six hours. This is due to meat between the teeth, which one cannot clean.

Tosfos (34b DH veha'Shelishi): Rashi's first Perush is correct. We discuss Chulin Al Taharas Terumah. Even though he eats it when he is Tamei, and the Torah forbids Chetzi Shi'ur, Chachamim were not stringent about Tum'as ha'Guf due to eating Tamei food.

Shiltei ha'Giborim (on Mordechai Berachos 48b:15): The Ge'onim say that if one forgot and blessed ha'Motzi before Havdalah, he should eat, lest his Berachah be l'Vatalah, and then make Havdalah.

Maharam (302, cited in Magen Avraham 271:12): If one blessed on bread before Havdalah, he eats, says Havdalah, and says ha'Motzi again after Havdalah. This is unlike saying 'feed the cattle', after which one need not bless again. There, it is a Torah Isur to eat before feeding one's animals.

Poskim

Rema (OC 271:5): If one blessed on bread before Havdalah on Motza'ei Shabbos, he should eat before (Havdalah).

Taz (7): He need not say ha'Motzi again after Havdalah.

Magen Avraham (12): The Rema permits to eat the slice of ha'Motzi. After this, he must interrupt to say Havdalah.

Magen Avraham (299:2): Tosfos (Shabbos 9b DH Leima) says that if one began eating b'Isur (after Shabbos, before Havdalah), one must interrupt (to say Havdalah). The Rema (271:5) connotes like this. They must hold that Havdalah is mid'Oraisa. According to the opinion that it is mid'Rabanan, if one washed before Havdalah, he makes Havdalah before eating. Since he has the entire night to say Havdalah, we are concerned lest he be negligent (and forget to say it). However, if he began eating, he need not interrupt.

Mishnah Berurah (271:26 and Sha'ar ha'Tziyon 32): The Magen Avraham says that according to the opinion that Havdalah is mid'Rabanan, if one said ha'Motzi before Havdalah, he need not interrupt to say Havdalah until he finishes eating. R. Akiva Eiger questioned this.

R. Akiva Eiger (271:5): The Rema did not say that he must interrupt for Havdalah. He holds that this is only according to the opinion that Havdalah is mid'Oraisa, therefore one who began when it was forbidden must interrupt. We hold that it is a Safek whether Havdalah is mid'Oraisa. Therefore, whether one must interrupt is a Safek about a mid'Rabanan law, so we are lenient.

Yechaveh Da'as (4 YD 41): If one ate meat, and a short time afterwards he blessed on something dairy, may he taste it to avoid a Berachah l'Vatalah? The Gemara (105a) says that one may not eat dairy until the next meal. The Rif, Rambam, Rosh, Rashba, Ritva, Me'iri and Ran obligate waiting six hours. The Mechaber rules like this, and this is the custom. However, Tosfos says that if one removed the table on which he ate and blessed Birkas ha'Mazon, he may eat dairy. Avi ha'Ezri, the Mordechai, Yere'im, Ba'al ha'Ma'or and Ro'oh agree. This is the custom of Chachamim of France and Narvona. The Rema says that the custom in Ashkenaz is to wait Sha'ah Achas (one hour, or a short time) after removing the table and blessing. Some say that even though we accepted the Mechaber's rulings, if one is slightly sick, it suffices to wait one hour. Since the Rambam (Teshuvah 84), Ge'onim and Shulchan Aruch (OC 215:4) hold that a Berachah l'Vatalah is a Torah Isur of Lo Sisa Es Shem Hash-m la'Shav, it is better to taste a drop of the food to avoid a Berachah l'Vatalah. This is even according to Tosfos (Rosh Hashanah 33a DH Ha) and others who hold that a Berachah l'Vatalah is only mid'Rabanan, and the verse is only an Asmachta. Some permit dairy within six hours of meat, but all forbid Berachah l'Vatalah at least due to an Asmachta from the Torah, so a Safek Isur is better than a Vadai Isur. The Rema (OC 271:5) says like this about one who blessed on bread before Havdalah. Also the Birkei Yosef and Shulchan Aruch ha'Rav say so. The Isur of Berachah l'Vatalah overrides the Isur of eating before Havdalah, for the former is more stringent. The same applies here.

Yechaveh Da'as: However, the Rambam and Tosfos hold that eating Isur is not considered eating. Even for an Isur mid'Rabanan, one does not bless before or after eating it. The Mechaber (OC 196:1) rules like this. If so, in any case a Berachah on a dairy food within six hours is l'Vatalah, for it is like a Berachah on Isur! However, that is only when the food itself is forbidden. Here, the food is Heter, just he eats it at a forbidden time. Even the Rambam and Shulchan Aruch agree that a Berachah helps in such a case. The Ramah says that one who eats Isur due to danger does not bless on it at all. The Bach (204) says that he agrees that if one must eat on Yom Kipur due to danger, he blesses. Perhaps here it is not considered total Heter. One must wait six hours due to fat from the meat sticks to the mouth, and the taste endures for six hours (Rashi), or due to pieces of meat between the teeth (Rambam). Mid'Rabanan, there is a mixture of Basar v'Chalav. Rather, we can justify the Berachah according to those who argue with the Rambam (regarding blessing on Isur), i.e. the Ra'avad, Rosh, Tosfos Yeshanim (Shabbos 23a in the margin) and Chut ha'Meshulash at the end of Tashbatz (3:29). Therefore, one should taste the milk due to a Sefek-Sefeka (two doubts). Perhaps the Halachah follows those who permit eating right after removing the table and Birkas ha'Mazon. And even if the Halachah is unlike them, perhaps it is like those who say to bless on Isur, and this evades a Berachah l'Vatalah, which is more severe. We can add the opinion of R. Klonimus (in the Mordechai Chulin 664). It seems that he permits Chetzi Shi'ur of an Isur mid'Rabanan. Tosfos (34b) says similarly. Pesach ha'Dvir supported R. Klonimus, but the Beis Yosef rejected him. Sedei Chemed (Ma'areches Ches, 16) distinguishes between Isurim mid'Rabanan with a source from Torah, and those without. For the latter, we do not say that Chachamim enacted like Torah law (Gitin 64b). Many Rishonim permit eating dairy l'Chatchilah right after removing the table and Birkas ha'Mazon, so this is no more stringent than an Isur mid'Rabanan without a source in Torah. Levushei Mordechai (YD 167) said to be stringent and not taste, for according to the Rambam it does not help, for it is still l'Vatalah. He did not consider everything I wrote here. Zechor l'Avraham (Basar v'Chalav) says that a case occurred, and he swallowed a little. Also, it was in winter, four hours after he ate meat. The Pri Chodosh says that in winter four hours suffices. Sedei Chemed says that this was a necessary part of the Heter; this is wrong. One should eat a little even if this is shortly after he blessed after the meat meal, like Shulchan Melachim says. The primary opinion forbids a Berachah l'Vatalah mid'Oraisa, so one should eat a little even if there is still meat between his teeth. Michtav l'Chizkiyahu (5:31a) brought from Piskei Rikanti that a case occurred in which one blessed on meat in the first nine days of Av, in which the custom is not to eat meat. He said that since many hold that Berachah l'Vatalah is only mid'Rabanan, he should not swallow any, even though it is a mere custom not to eat meat then. However, Sedei Chemed cites Piskei Rikanti to say oppositely. There is no Simchah in swallowing a tiny amount. Be'er Moshe (4:24) said that one who blessed on milk within six hours of eating meat should swallow some.

Teshuvos v'Hanhagos (2:141): If one drinks a drop of whiskey without pleasure (just to please the host), he may not bless on it.

Note: According to this, when one tastes a little to avoid a Berachah l'Vatalah, he must taste enough to get some benefit from it.