Answer: The Rema (OC 301:46) writes that as one is not allowed to squeeze clothes out on Shabbos (sechita), wet clothes are muktze on Shabbos. The Mishna Berura (308:63) adds that clothes that are wet when Shabbos came in are muktze the entire day.
Nonetheless, the Shulchan Aruch (OC 301:19) writes that something which will automatically become usable does not become muktze during bein hashmashos. R’ Yitzchak Yaakov Weiss (Minchas Yitzchak 1:81) explains that the Mishna Berura may be referring to clothes that one isn’t sure would dry over Shabbos. As the owner clearly never intended on using them over Shabbos, they are muktze. Thus, R’ Shmuel Wosner (Shevet Halevi 1:62:3) writes that if one was sure that clothes will dry on Shabbos, they are not muktze (See Shemiras Shabbos Kehilchasa 15:15).
R’ Shlomo Zalman Auerbach (Minchas Shlomo 1:10:7) and R’ Moshe Feinstein (Igros Moshe OC 5:22:38) write that if it wasn’t for the noise, one would be allowed to put the clothes in on Friday and let the dryer finish on Shabbos. Providing the machine ended before Shabbos, however, one may remove the clothes from a dryer on Shabbos.
In conclusion, one may remove dry clothes from a washing line or dryer to wear on Shabbos, even if they were wet when Shabbos came in.