What are the differences (if they exist) in meaning between the two questions?
"Do you remember" is mostlty used when referring to spontaneous memories, very often with nostalgic overtones. For instance "Darling, do remember that little hotel we stayed in by Lake Como? Wasn't it romantic? or "Mum, do you remember Dad getting his fish and chips grabbed by that seagull?"
"Can you remember?" is more often used for specific pieces of information which require concious recall. For instance "Can you remember what we did with the suitcases?" or "Can you remember the way to work out the circumference of a circle?"
The two forms are sometimes combined as in "Darling, do you remember that romantic little hotel near Lake Como? Can you remember what it was called?"
These aren't hard and fast rules, either form can be used for either purpose but there is a definite tendency to use them in the way I've indicated.
Can you remember suggests that you are not sure of what you are asking. For example: "Joe, can you remember if we shut the doors?" In that sentence, the person asking is not sure if he shut the doors.
On the contrary, Do you remember suggests that you know what you are saying, but want to find out if the other person knows about it. For example: "Joe, do you remember that we shut the doors?" In that sentence, the person asking knows that he shut the doors, but he wants to know if Joe remembers, perhaps to prove a point.