Is 不要在厨房里留下任何废物 the correct way of saying "do not leave any waste in the kitchen"?
不要在廚房裡留下任何廢物


It is an acceptable translation of "do not leave any waste in the kitchen", you can use it just fine.

However, if you want to be a little more formal, you can write "please don't leave garbage in the kitchen" (請勿在厨房留下垃圾)

you don't need '裡' in '在廚房裡' ;  '在' alone in '在厨房' already covered the meaning of 'in' 

you can remove '任何' in '任何垃圾' ;  just like 'dog' in 'no dog allowed' already meant 'all / any dog'

Shorter note get the point across better. Just like '不准泊車' is better than '各位先生女士，在任何時間內都千萬不要在這個地點停泊任何車輛，包括摩托車及電動輪椅' (you can google translate this sentence) -- No one would read it.