Both of these questions are correct. However, they mean two very different things:
Do you remember to study English? = Do you study English because you remember to do it?
Do you remember studying English? = Do you have the memory of studying English at some time in the past?
These questions are examples of a few verbs that can take both the gerund (doing) or the infinitive (to do), but change meaning depending on which form. Other verbs can only take the infinitive OR the gerund. If you are unsure of which verbs take which form, use this interactive gerund or infinitive tool to test yourself and improve your knowledge.