Well, when you UPDATE a row, it will only change the values you want. So if you don't change the value in the query, it will stay the same.
If you want the value to be 0000-00-00, then just define it that way:
$date = (!empty($_POST['date']))?$_POST['date']:'0000-00-00';
Then, in your query just use $date to insert the value into that column.
~Brett