Answer: There are a few potential Halachic issues to be aware of. R’ Moshe Feinstein (Igros Moshe OC 1:112) and R’ Yitzchak Yaakov Weiss (Minchas Yitzchak 3:48; 50) write that one must not brush one’s teeth with regular toothpaste on Shabbos, as smoothing the toothpaste onto one’s teeth is the transgression of memareiach, smoothing out a solid (or semi-solid) object, as well as sechita, squeezing the liquid out of the bristles.
However, R’ Ovadia Yosef (Yabia Omer OC 4:27:2; 28) disagrees, maintaining that the memareiach only applies when one leaves a layer of paste and not when it gets brushed off immediately. Nonetheless, R’ Eliezer Waldenberg (Tzitz Eliezer 7:30) challenges this, explaining that one purposely rubs the toothpaste into one’s gums and teeth. Even if it gets washed away soon after, one has transgressed the issur of memareiach.
R' Yechiel Yaakov Weinberg (Seridei Eish 30) explains that as the water is immediately discarded, there is no issue of sechita. Unlike washing clothes where the water is added and extracted in order to clean them, the brush is not being cleaned by the water.
R’ Eliezer Waldenberg suggests that brushing teeth is also an issur of molid, forming a new substance, as one changes the solid consistency of the paste into a liquid (See Rema OC 326:10). However, R’ Ovadia Yosef (Yabia Omer OC 4:30) writes that this is not nolad as the paste isn’t a proper solid to begin with.
R’ Yitzchak Yaakov Weiss and R’ Eliezer Waldenberg write that another issue is that brushing frequently causes one’s gums to bleed. R’ Ovadia Yosef writes that people know whether this is likely when they do so. If it rarely happens, one does not need to be concerned.
R’ Shlomo Zalman Auerbach (Minchas Shlomo 2:35:3; Comments to Seridei Eish ibid.) writes that while there are strong challenges against the prohibition, the accepted practice is to avoid brushing one’s teeth in the usual manner on Shabbos.
In conclusion, many Sefardim follow R’ Ovadia Yosef and brush their teeth normally on Shabbos. Ashkenazim should ideally use either a Shabbos Toothbrush or a dry soft brush, that avoids the aforementioned issues.