May one dry himself with a towel on shabbos after using the mikvah? If not, why not? If yes, why?
According to Rama 302:10, you may clean yourself with a towel, based on Derech Lichluch:
This based on another concept called “derech lichluch”, which means that the water is applied in a dirtying manner. When drying hands on a towel, the towel does not become cleaner; on the contrary, it gets dirty. So even though water is coming in contact with the towel, since it is not cleaning the towel, it is permitted. The same rule applies to wiping up spilled water on the floor: even though the water is being absorbed into the towel, since it is done in a “dirtying manner”, it is permitted
Quote obtained from Footnote 8, here
As discussed here a major issue is using the towel to dry hair, with opinions ranging from permitted (as long as the towel absorbs the water completely) to completely forbidden, with the in-between view of allowing gentle patting.
The Vilna Gaon would not go to the Mikvah at all on Shabbos to avoid this problem, even though he would generally go during the week.
I've heard of some Chassidim (of those groups that originate on Poland) not drying with a towel after the Mikvah not because of Shabbos, but because they wanted to recognize the preciousness of the water, so you may have seen that, but I don't think there is any Halachic concern that motivates that practice.