kcott raised the most important points. To expand on that a bit...

The first time you execute an SQL statement like this:

update table set ID = 10 where ID = 5"
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that one execution will cause every row with ID=5 to updated to ID=10. If you execute the exact same statement a second time (e.g. trying to set ID=11 instead of 10), nothing will happen, because you no longer have any rows with ID=5 (all those rows now have ID=10).

I would suggest using a SELECT statement to get all the rows having ID=5, and then for each row returned by that query, do an update with desired new value for ID:

my $update_sth = $dbh->prepare( "update table set ID=? where ID=5 and 
+Name=?");
my $select_sth = $dbh->prepare( "select Name from table where ID=5" );

my $new_ID = 10;
$select->execute;

while ( my $row = $select->fetchrow_arrayref ) {
    my $name = $$row[0];
    $update_sth->execute( $new_ID++, $name );
}
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(You should probably add some error checking, to make sure the updates work as intended.)

Updated to fix a typo in the 2nd line of the code snippet (was "IS=5") -- thanks kcott!