"Laat je de kleren daar liggen?"
Translation:Are you leaving the clothes lying there?
I typed "are you letting the clothes lay there?" and it was marked as wrong, and that the correct word would have been "lie" instead of "lay," but if we're going to get into the weeds about grammar, "lie" is incorrect, and "lay" is correct. "Lay always requires a direct object (the clothes) whereas "lie" does not. A person lies down. Or at least that's how I understand it.
You're right and wrong: completely right about what you say about 'lay' needing a direct object, whereas 'lie' doesn't. (But 'lay' can be the past tense of 'lie' here, which isn't the case here.) As letsrockltd says, the clothes aren't the direct object of 'lay/lie'. The sentence is really 'Are you letting the clothes to lay/to lie there, but in English we omit the 'to' part of the infinitive here. 'The clothes' is the direct object of the verb 'are letting'. But the clothes are being left to lay/to lie. They aren't laying anything themselves -- you need 'to lie' because you need a verb that doesn't need a direct object (an intransitive verb). Sorry if that all sounds enormously complex...