With regard to using toothpaste, I believe Rav Moshe Feinstein ruled that there was a concern of memareyach (smoothing) though i think I heard that Rav Yisrael Belsky said he was misled regarding the physical reality and that this should not in fact be a concern since there is no significant residue that remains on the teeth. Additional concerns have been raised regarding cleaning the brush (libun) or extracting liquid from the brush (dosh) though these concerns also are at best controversial. I believe Rav Yosef Dov Soloveitchik is quoted as being lenient regarding brushing one's teeth with a toothbrush with or without toothpaste.

With regard to using toothpaste, I believe Rav Moshe Feinstein ruled that there was a concern of memareyach (smoothing) though i think I heard that Rav Yisrael Belsky said he was misled regarding the physical reality and that this should not in fact be a concern since there is no significant residue that remains on the teeth. Additional concerns have been raised regarding cleaning the brush (libun) or extracting liquid from the brush (dosh) though these concerns also are at best controversial. I believe Rav Yosef Dov Soloveitchik is quoted as being lenient regarding brushing one's teeth with a toothbrush with or without toothpaste. See also: http://halachipedia.com/index.php?title=Brushing_Teeth_on_Shabbat