Q. Can one brush his teeth on a Ta'anis without using any water or toothpaste, just one's own saliva and then spiting it?
A. Mishna Berura (567: 11) rules that when in pain and discomfort one may wash the mouth with water, while being careful to keep one's mouth down to ensure no water is swallowed. The above is permitted on the a regular Taanis Tzibur. Mishna Berura is more stringent on Tisha Beav when he permits only in a case of severe pain and discomfort, while he prohibits completely on Yom Kippur. See also Minchas Yitzchok 4: 109 and Kovetz Halochos 2: 9 that are stringent, However Aruch Hashulchan (567: 3) is lenient.
On question 2380 regarding if one may store and save his saliva for use during Yom Kippur to swallow necessary medical pills, we wrote:
"Mishna Berura (567: 13), quotes Magen Avrohom that it is permitted to swallow our owns saliva, even though we rule that it is not permitted to drink even liquids that are not fit to drink, swallowing saliva is different, since the intention is not for drinking at all.
Bach (612). cites different opinions regarding how much saliva is one allowed to swallow in Yom Kippur. Chayei Adam (132: 22) and Mateh Ephraim (ibid. 22) maintain that on the onset one should preferably avoid swallowing saliva that has accumulated in the mouth.
Moadim Uzemanim (1: 59) mentions that Hagra’ch Soloveitchik and the Brisker Rov zt’l were extremely careful not to swallow their own saliva when it accumulated during Yom Kippur.
However, many Poskim are lenient and maintain that this is not the accepted tradition. (Beis Meir, Eishel Avrohom Butchatch; 613. Aruch Hasulchan ibid. 4, Piskei Teshuvos 567: 2). Excepting, when particles of food are still in one’s mouth. at the beginning of the taanis.
Moadim Uzemanim (ibid.), adds that on Yom Kippur, one should not intentionally accumulate saliva in one’s mouth and then swallow it.
Horav Shlomo Miller’s Shlit’a opinion is that when in need for swallowing necessary pills, it is permitted to store one’s saliva even in a cup and save it for use during Yom Kippur.
The Rov's opinion is that certainly on the lesser Ta'anesim it is permitted for one to brush his teeth without using any water or toothpaste, just one's own saliva and then spiting it.
Rabbi A. Bartfeld as advised by Horav Shlomo Miller and Horav Aharon Miller Shlit’a
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