Close.
last OUTER;
should be
last OUTER unless $line = <$FH>;
You only want to do it when the end of the file is reached.

Update: Fixed these problems: $FH indicated a file handle but contained a filename, $FH was used as a file handle even though FILE was the file handle, $batch was not initialized, and $line was uselessly checked for "go" twice.

foreach my $filename (@files) {
   open(my $fh, '<', $filename)
   or die("Unable to open input file $filename: $!\n");

   my $batch = '';

   OUTER: while (my $line = <$fh>) {
      while (not $line =~ m/^\s*go\s*$/i) {
         $batch .= $line;
         last OUTER unless $line = <$fh>;
      }

      $dbh->do($batch);
   }
}
[download]

or without nested loops:

foreach my $filename (@files) {
   open(my $fh, '<', $filename)
   or die("Unable to open input file $filename: $!\n");

   my $batch = '';

   while (my $line = <$fh>) {
      if (not $line =~ m/^\s*go\s*$/i) {
         $batch .= $line;
         next;  # Go read next line.
      }

      $dbh->do($batch);

      $batch = '';
   }
}
[download]