I find it difficult to go a day without brushing my teeth. Is it okay to brush my teeth on a Ta'anis (fast day)? I imagine it's forbidden on Yom Kippur, but what about the other fasts?
According to this article, most Poskim do allow one to brush one's teeth on the three minor fasts, especially if they are going to be in some level of discomfort.
..While the Kaf Hachaim (OC 567:13) forbids rinsing even with less than a revi’is, most poskim are more lenient, especially when one is uncomfortable. Thus, the Rema (Darchei Moshe OC 567:2), Elya Rabba (567:5) and Baer Heitev (OC 567:5) relate that the Maharil used to rinse his mouth with water while being particular not to swallow any.
The Mishna Berura (567:11 quoting the Chayei Adam 132:20) writes that on one of the minor fasts one may rinse their mouth with water if they are uncomfortable and on Tisha B’av, one may only do so if they are suffering. One doing so should bend over the sink to ensure that they don’t swallow any water. A healthy person cannot rinse one’s mouth on Yom Kippur.
The poskim write that brushing one’s teeth is the same as rinsing one’s mouth (See Minchas Yitzchak 4:109).
R’ Ovadia Yosef (Yalkut Yosef, Ta’anis, 13) writes that if one is uncomfortable not brushing one’s teeth, one may do so, though ensure that one uses less than a revi’is of water and try not to swallow any.
The Aruch Hashulchan (OC 567:3) is more lenient still, allowing one to rinse (but not gargle) even with more than a revi’is of water.
In conclusion, one who wants to brush their teeth on a minor ta’anis may do so, though they should ensure that they don’t swallow any water.