Removing only birth years from a group in my Address Book
Ok.
   Since you have your group already you group.

set targetGroup to "BD" 
tell application "Contacts"
    set thePeople to people of group targetGroup
    repeat with i from 1 to number of items in thePeople
        set this_item to item i of thePeople
        set thisDate to birth date of this_item
        set year of thisDate to 1604

        set birth date of this_item to thisDate
        save
    end repeat


end tell


I tried to set the year to "" . But it changes the whole date. day and month to something else.

So I checked what a yearless birth date 'date' looked like and they all had 1604 in the year position.

setting the year to this seems to work.