Beitzah, chapter two, mishnah 3. In the previous mishnah, Beis Shammai and Beis Hillel disagreed about whether a person may immerse himself on Shabbos. Our mishnah opens specifically with a case in which the two houses are in agreement and concede to one another.
"Ushavin shemashikin es hamayim bichli even":
The word mashikin means to make things kiss. When a person had water that was tamei and wished to purify it, it was enough for him to bring it into contact with the water of a mikvah. The contact joins the two bodies of water, and the water that had been tamei until then becomes tahor. This is the law of hashakah. Even if the waters barely touch one another, that is sufficient.
The mishnah is precise in stating that the hashakah is done in a stone vessel, because a stone vessel is not susceptible to tumah. There is therefore no immersion of a vessel here at all, since a stone vessel has no status with regard to tumah and is always tahor, and immersion accomplishes nothing for it. The water itself may be purified on Yom Tov and on Shabbos, and there is no problem with that.
"Aval lo matbilin":
One may not place the water into a wooden vessel, for example, since a wooden vessel is susceptible to tumah. And we need not be speaking specifically of a wooden vessel that was already tamei beforehand: from the moment the tamei water is placed into the wooden vessel, even before the immersion, the water renders the vessel tamei. And once the wooden vessel has become tamei, it certainly may not be brought into the mikvah, for even though the main intention is to purify the water, in practice the vessel is purified at that very same moment as well.
Two additional cases of permitted immersion:
"Umatbilin migav l'gav" - a vessel that has already been immersed and brought into a mikvah, but the immersion was done for the sake of chullin, for ordinary needs that are neither Terumah nor kodashim, and now its owner wishes to purify it for the sake of Terumah; or it was purified for the sake of Terumah, and now he wishes to use it for kodashim, for sacrifices. Such use requires an additional immersion, and since this is nothing more than an additional immersion, it is permitted on Shabbos and Yom Tov. It does not look like a vessel that broke and is now being repaired, the way it looks with a vessel that is tamei. Rather, the vessel is simply not tahor enough for the new use, and now it is being raised from one level of purity to the next.
"Umeichavurah l'chavurah" - one who immersed his vessels in order to eat the korban Pesach with one group, and now wishes to eat with a different group. Strictly speaking, no additional immersion is needed here at all, but if he wished to immerse again out of an added stringency, he may certainly do so on Shabbos and Yom Tov, since the immersion is not required in the first place and his entire intention is nothing more than an added measure of purity.
In summary: Beis Shammai and Beis Hillel agree that one may bring tamei water into contact with the water of a mikvah in a stone vessel, which is not susceptible to tumah at all. But one does not immerse the water in a vessel that is susceptible to tumah, such as a wooden vessel, since the water renders it tamei and he turns out to be immersing a tamei vessel. So too, immersion from one level of purity to another and immersion from one group to another are permitted, since they involve no repair of a tamei vessel but only a raising of the level of purity.