Yes, you can hide empty custom contact fields in each field group on a Contact's profile page.
To do so, navigate to the Contact's profile page. Then, click the "Hide Empty Fields" option at the top of a field group. All empty fields for that specific group will be hidden from view.
To show empty custom contact fields for that group at any time, click "Show Empty Fields."
Note that this option will only hide empty custom contact fields. This will not hide empty default fields (i.e., Phone and Organization).