Enter a birthdate for a contact via "Edit Contact". Then load the contact record in the edit form for sample profile gid=1 (you can use civicrm/profile/edit?gid=1&reset=1 if you're working with the contact record linked to your logged in user). Save the profile form. The contact record's birth_date will be set to NULL.
NOTE: I checked all (most) of the other contact record fields to see if other values which are NOT in the profile form (e.g. deceased_date, website, nickname, prefix ...) are affected. Based on my testing, birth_date is the only one that is "unintentionally" set to NULL. However, when you figure out where this is happening, perhaps you'll see other fields affected ??