Using the dataset filter property will not work as you want since it doesn't accept LIKE.
You could use a TQuery component and set its SQL property to this:
select * from Table
where Expiration like '%/' + :YEAR
In your form, you should add this code:
procedure TForm1.Button1Click(Sender: TObject);
begin
Query1.Close;
Query1.ParamByName('YEAR').AsString := Edit1.Text;
Query1.Open;
end;