Mishnah 3 presents us with a dilemma. A person who bakes matzos on Pesach - and this was the practice in ancient times, to bake matzos on Pesach itself - is obligated, like anyone who bakes bread, to separate challah and give it to a kohen. The Mishnah asks what the law is in a case where the dough became tamei: once the dough has become tamei, the challah separated from it is also tamei, and challah that is tamei is not fit to be eaten by a kohen. And here, on Yom Tov, a problem arises: what is a person to do with that challah?
The problem on Yom Tov of Pesach:
Baking: If it were not Yom Tov, one could bake the tamei challah and thereby prevent it from becoming chametz. But it is forbidden to bake on Yom Tov without need, and the tamei challah is not fit for anyone to eat - so this option is not open to him.
Destroying it: He may not feed the challah to a kohen's dog, since it is forbidden to destroy consecrated items on Yom Tov; he must wait until the festival ends.
Leaving it as it is: He cannot even leave it lying there, since it is bound to become chametz. If it were not Pesach, he could leave it as it is, but since this is Yom Tov of Pesach he is caught in a bind.
From here the Mishnah opens with the question: "Keitzad mafrishin challah min ha'isah kesheh'hi teme'ah b'Yom Tov?" - How does one separate challah from dough when it is tamei on Yom Tov?
Three opinions in the Mishnah:
Rabbi Eliezer: One should not designate it as challah until after the baking. That is, he should not separate the challah beforehand; rather, he bakes everything, and only after the baking does he designate one portion as challah. Ordinary bread, which is not challah, does not require taharah and may be tamei, and therefore it is permitted to bake it on Yom Tov for the purpose of eating. This opinion rests on Rabbi Eliezer's position that one may separate challah after the baking, when the loaves are placed in a basket and the basket joins them together for challah.
Rabbi Yehudah ben Beseirah: He should place it in cold water, and the cold water will prevent it from becoming chametz.
Rabbi Yehoshua: In his view there is no problem here to begin with. He separates the challah and does not bake it, and even though it will become chametz - "lo zehu chametz shemuzharim alav b'bal yera'eh u'bal yimatzei" - this is not the chametz regarding which one is warned with "it shall not be seen" and "it shall not be found." For the verse says, "and no chametz shall be seen belonging to you," and the prohibition of seeing applies to your chametz; but this chametz is destined to be given to a kohen: it does not belong to any particular kohen, and you are also not responsible for it. Rather, "mafrishtah u'manichatah ad ha'erev" - he separates the challah and leaves it until evening, "v'im hechmitzah hechmitzah" - and if it became chametz, it became chametz.
The roots of the dispute:
Rabbi Eliezer does not accept Rabbi Yehudah ben Beseirah's approach, because it is not a perfect solution and the dough may still become chametz.
Rabbi Eliezer likewise does not accept Rabbi Yehoshua's approach, and as the Gemara explains: when a person separates challah or Terumah, his separation has the status of a vow, and it can be brought before a sage and annulled. And since it is in his power to nullify the status of the challah, it would then revert to being his - so he is liable to transgress the prohibition of owning chametz. For this reason the practice is not permitted.
On the other side, Rabbi Yehudah ben Beseirah does not accept Rabbi Eliezer's approach, since in his view there is no permission to bake the entire dough, including the portion that is destined to become tamei challah which is not meant to be baked. A person may not bake the whole batch when part of it has no need for baking at all.
Rabbi Yehoshua, for his part, is not troubled by Rabbi Eliezer's concern, which Rabbi Yehudah ben Beseirah also accepts, that he may have his vow annulled. In his view, the concern exists only if he actually goes and annuls the vow; but as long as he has not annulled it, there is no problem, and the chametz is not considered his so as to transgress "it shall not be seen" and "it shall not be found."
In summary: Our Mishnah discusses separating challah from tamei dough on Yom Tov of Pesach, when it cannot be baked, cannot be destroyed, and cannot be left to become chametz. Rabbi Eliezer instructs that the designation be delayed until after the baking, since the basket joins the loaves together; Rabbi Yehudah ben Beseirah instructs that it be placed in cold water; and Rabbi Yehoshua rules that this is not the chametz regarding which one is warned, and therefore he separates it and leaves it until evening, and if it became chametz, it became chametz. The dispute revolves around the quality of each solution: the concern that it will become chametz even in cold water, the baking of a portion that does not require baking, and the possibility of having the separation annulled and thereby returning the chametz to his ownership.