You may not wash or hang up wet laundry on Shabbat (or Jewish festivals).  The halachot for drying laundry depend on whether you use a clothesline or a dryer:

Shabbat: Laundry: Clothesline
You may take down laundry on Shabbat only if it was dry before sunset on Friday, and only if you don't:

• Transfer the laundry from one halachic domain to another (hotza'a), or
• Give the impression that the laundry had been washed on Shabbat (mar'it ayin).

If laundry on a clothesline is still wet at sunset on Friday, the laundry is muktza and you may not take it down or use it during Shabbat. This is different from the case of a dryer.
Reason On the clothesline, there is no certainty that the laundry will dry during Shabbat (it might rain, it might be cold or cloudy...), so the person cannot have in mind that it will dry during Shabbat.

Shabbat: Laundry: Dryer
Laundry in a dryer (even if it was wet at sunset) that was turned on before sunset on Friday (or Jewish festivals) is not muktza, even if you do not intend to wear it.  You may remove the dry laundry from the dryer on Shabbat as long as no light goes on.