Here's one way:

# ...
$dbh->{RaiseError}=1;
$dbh->{PrintError}=0;
# ...
for my $stmt( split /;\n+/, join('',<FH>) ){
    my $sth = $dbh->prepare($stmt);
    $sth->execute;
    fetch_and_display($sth) if $sth->{NUM_OF_FIELDS}
}
[download]

NUM_OF_FIELDS should reliably mark statements like SELECT that return fetchable data so this does a bit more than simply looping over $dbh->do().

For more complex stuff I do things like use SQL comments to give names to the statements and/or include a trailing square-bracket list of bind values before the semicolon for placeholder statements

-- get_customer_name
SELECT name FROM customer WHERE id = ? [47];
[download]

You can use DBI::Shell in a program rather than a shell (I've done it, it required a minor subclass IIRC). At the moment DBI::Shell AFAIK uses something very similar to what I've shown above so you won't really gain anything that way.