If you push the cabinet with your hip, does the gap close up? If the gap closes when pushed-in, the cabinet is confirmed to be loose.
What comes to mind is to remove the drawers and secure the cabinet to the back and side walls from inside the cabinet. It could be something as simple as tightening existing screws.
It is not possible to say what could happen if you ignore the condition. Maybe nothing, maybe something bad.
Does your side wall have the beveled mirror? If so, you may run a risk of the counter-top hammering the bottom edge of the mirror of which is naturally a bad condition.