last OUTER unless defined($line_B = <FILE>); # this should do
[download]

But beside that, you have more problems. You're not checking the return value of 'open', the file might not open. Furthermove, if you do:

     while ((my $line_B = <FILE>)) {
         last OUTER unless defined ($line_B = <FILE>)
[download]

Why would I be skipping every odd line?

Because you read a line in the while, and then discard that line, and read the next line in your unless statement.
Perl --((8:>*

In line 10, the one you note, $FH is the file name, not a handle. I'm not sure you want that line at all, since you just read a line to the same variable in the while condition. I suspect you're trying to detect eof there, but you'll also clobber your loop variable and skip every other line of the file.

I think you need to review your code carefully. I'm not sure what you're trying to do, but I'm certain that this won't do it. Your do {} until //; loop keeps tacking the same text onto $batch until it matches something that that it never will if it didn't the first time round. You'll loop endlessly and $batch will grow until you run out of memory.

Update: If you are trying to detect eof to skip the last line, this works,

while (<$fh>) {
    last if eof;
    print;
}
[download]

To answer your question, anyway, on the last line of your input file the file itself gets closed, so you can operate on that $line, but you can't read further lines. What did you expect to do?!? Update: ignore this, I didn't notice your typo. Right answer e.g. at Skeeve's reply.

Compare and contrast:

# use this form of foreach instead of 'my $FH = $_'
foreach my $FH (@files) {
    
    # use 3-arg form of open() for safety, and deal with
    # being unable to open the file.
    open my $FILE, '<', $FH or die("Cannot read $FH: $!");

    # deal with each line of the file.  while loop exits
    # when EOF is reached.
    my $batch;
    OUTER: while (<FILE>) {
        $batch .= $_; # cat this line to $batch

        # if we just read a line that's 'GO', run the batch and
        # reset the variable $batch to be empty
        if (m/^\s*go\s*$/i) {
            $dbh->do($batch);
            # you should check for and deal with DBI errors here
            $batch = '';
        }
    }

    # if we didn't reach a 'GO', there will be stuff in $batch, so
    # we should report that fact.
    warn "File '$FH' contained unexecuted commands:\n$batch\n";
}
[download]

Once you get that down, it becomes much easier to explore other algorithms, like this one:

foreach my $FH (@files) {
    open my $FILE, '<', $FH or die("Cannot read $FH: $!");
    my $data = join('', <$FILE>); # slurp file!

    # split on lines containing only 'GO' (and optionally, whitespace)
    foreach my $batch ( split /\n\s*go\s*\n/i, $data ) {
        # if you need the 'go', you'll need to do $batch.="\nGO"; here
        $dbh->do($batch);
        # error handling here
    }
}
[download]

Because I very rarely program in Perl.

No mater what the language you should always check the return value from system and library calls.