As AppleCare doesn't cover accidental damage, Apple is under no obligation to make repairs free of charge to machines that are physically damaged or otherwise by the user. However, even if it is obvious that the machine has been dropped, they would still be prepared to fix it for a fee (which can often be prohibitive, depending on the severity of the damage). However, it would generally be far cheaper to go elsewhere for charged repairs.
If the machine works but is simply aesthetically damaged, I expect their answer would definitely be along the lines of "live with it" or "pay us lots of money to put it right."
Worth a try, as always, first though.