You'll need to set local $dbh->{RaiseError} = 1; above your while-loop for that to guarantee working in all situations. And, since you're using local, I'd recommend a bare-block.

my $q   = 'INSERT INTO some_table (data) VALUES (?)';
my $sth = $dbh->prepare($q);
{
    local $dbh->{RaiseError} = 1;
    while(my $data = $csv->get_next_line()) {
        eval{
            $sth->execute($data);
        }

        if($@) {
            next;
        }

        # More code here...
    }
}
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Of course, with RaiseError set to false, you can also do the more readable:

while ( my $data = ... ) {
    $sth->execute( $data )
        or next;
}
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It's up to you and the style you choose to work with. Either way, be consistent. (You wrap your execute(), but not your prepare() ... tsktsk!)