The criterion for things which invalidate the fast is when a substance reaches the body cavity through an open passageway, such as the mouth or nose. Hence, if what the asthma patient uses is for relief only, his fast is not invalidated. But if a substance exits from the aspirator, such as a powder or water spray, and enters the body cavity, then it invalidates the fast.
However, if the patient needs to use it, then his illness excuses him from the fast, and he makes up the missed day. And if he does not need to use it, or can delay it until after sunset, then it is unlawful for him to use it in the absence of any need.