I would say that "burnable trash" and "non-burnable" trash are the most natural (for me). The meanings themselves are the same as "combustible" and "non-combustible".

Best,

Chris

Word usage has a lot to do with your discretion according to the context of what you are translating. If you are using the adjective "combustible" or "noncombustible" in your work, it is more likely that you will see the word "waste" after it, rather than the other two. However, there is also the point that "waste" can be interpreted in broader terms, while there is a certian degree of specificity when talking about "garbage," or "trash." It's up to you to decide from the context of your work, but I feel that "waste" would probably be the proper terminology.