"Dry up" is something one does to a liquid, or the liquid does on its own. "The lake dried up" or "The sun dried up the puddles."
"Dry" is something one does to something with liquid on/in it, or the object does on its own. "The plates dried slowly" or "I dried the dishes."
I don't think I'd include the up.We don't need to dry up the dishes in our house