I think the confusion comes from the way that the students are tested when learning English as a foreign language. If they stick to the general rule of using fast as an adjective and quickly as an adverb, they are unlikely to make a mistake. Although fast CAN be used as both an adjective and an adverb, it is not ALWAYS the case, so if a student is learning academic English in order to pass exams, they are often encouraged to stick to this rule.
For example, in the following gap-fill exercise:
Sometimes an allergy can disappear as ____________ as it arrived, without any treatment.
The answer key only mentions "quickly/suddenly".
"Fast" would be incorrect.
Even if "disappear as fast" sounds perfectly acceptable to some native English speakers, this information is not very helpful to a foreign student who is trying to pass an exam.
I guess the best advice to give is that "fast" can be both an adverb and an adjective, but when in doubt, use "quickly".
