Actually clothing protects your skin from being burned, but that's real life and in places that arent hot enough to instantly set everything on fire, lol.
That’s actually a good question. I never thought of trying that before... However, you do still need to wear the Zora tunic to convince Muzu to cooperate with you, as well as for Sidon to bring you in to the fight, so whether you can or not, you can’t do the Vah Ruta campaign naked.
You can complete the entire game without wearing anything. You can get everything you really need from food boosts. I personally wouldn’t though just cuz I’m a huge fashionista and I like my clothes.
Actually, I guess three of the Divine Beast campaigns require certain armors to be worn. But...