What should one do to clear sweat or water from one’s mustache or beard, or eyebrows by hand on Shabbos?
We do not squeeze hair on Shabbos, therefore it can not be cleared using one’s hand, because it will be squeezing the water out, which is sechita on Shabbos ( others say that it would be similar to melaben- laundering). The best idea would be place a napkin or towel on top of the hairs, so the towel will absorb the water on it’s own without it getting squeezed from between the hairs. See sources.
Rambam (Shabbos) 9-10, Magen Avrohom 320- 23, M:B 320-55, 326-21, 330-7 Biur Halacha 302-9 D:H Assur, Igros Moshe O:CH 1-133. Regarding using a towel there are different opinions, but most poskim agree that it is permitted to place it lightly, or rub it softly on his head, Toras Shabbos 320-23, Minchas Shabbos Kometz Hamincha 86-6, Shmiras Shabbos Khilchoso 14-22 ftnt. 68, Shulchan Shlomo 320 10-2, 3, ftnt.26, Kaf Hachayim 320-111 from Ben Ish Chai, Orchos Shabbos 13 ftnt. 89 in the name of R’ Eliyashiv zt”l, Chut Shani Shabbos 2 pg. 62, Oz NIdberu 1-55, Meohr Hashabbos 3 pg. 369