Question: What can one do if they get soaked in the rain over Shabbos? Can they stand next to the radiator and hang their wet clothes up?
Answer: The Shulchan Aruch (OC 301:46) writes that one can’t dry wet clothes next to a fire on Shabbos. The Magen Avraham explains that it is forbidden both because of bishul (cooking or heating) as well as melaben (whitening or improving the colour).
Thus, the Mishna Berura (301:169) writes that one can’t stand next to a heater while wearing wet clothes. This only applies if the water could get heated up to yad soledes bo (approx. 43 °C, See Igros Moshe OC 4:74).
The Mishna (Shabbos 146b) writes that it is prohibited miderabanan to lay out wet clothes to dry on Shabbos. Rashi explains that it will give the impression that it is permissible to wash them on Shabbos. Thus, the Shulchan Aruch (OC 301:45) writes that one mustn’t hang wet clothes up normally to dry even where others won’t be able to see them.
R’ Yehoshua Neuwirth (Shemiras Shabbos Kehilchasa 15:35; n119) writes that one may, however, hang up a wet raincoat as everyone will realize that it became wet in the rain. Similarly, R’ Dovid Ribiat (The 39 Melochos, p719) writes that this prohibition doesn’t apply to clothes that aren’t normally washed, as people won’t suspect they’ve been washed. Thus, one may hang up a jacket that is normally dry-cleaned.