When we say "verb the object adjective", we mean "make the object adjective by verbing". "Boil the kettle dry" means "make the kettle dry by boiling", just as "paint the door red" means "make the door red by painting".
When we use an adverb, like "verb the object adverb", we mean "do the verb in an adverb way". In this case, the sentence almost certainly means "make the kettle dry by boiling", because the other sense would be unclear. ("Boil the kettle in a dry way"? What does that even mean?)
This is sometimes complicated, because some words can be both adjectives and adverbs (like tight), and sometimes people use adjectives where they should use adverbs, especially when speaking informally.