Some improvements. Notes:

• %ignore now holds compiled patterns.
• Counts are now in %ignored.
• glob2pat should be faster by processing multiple characters at once, but that shouldn't affect your run time.
• Removed $do_not_print to reduce clutter. A label is used on the while loop instead.
• In the second snippet, Regexp::Assemble is used to factor out the "^" and "$" which is common to all ignore regexps.
• The chomp is probably unnecessary.
• If you only use $col[0] of @cols, replace
  my( @cols ) = split(" ", $_);
  with
  my( $col0 ) = /^\s*(\S+)/;

{
   my %patmap = (
      '*' => '.*',
      '?' => '.',
      '[' => '[',
      ']' => ']',
   );

   my ($norm) = map qr/[^$_]/, join '', keys %patmap;

   sub glob2pat {
      my $globstr = @_ ? $_[0] : $_;
      $globstr =~ s{($norm+|.)} { $patmap{$1} || "\Q$1" }sge;
      return "^$globstr$";
   }
}

my %ignore = map { $_ => qr/$_/ }, map glob2pat, @list_regexps;
my %ignored;

LINE:
while(<FILE>) {
   chomp;
   my( @cols ) = split(" ", $_);

   ...

   for my $re_name ( keys %ignore ) {
      my $re = $ignore{$re_name};
      if ( $cols[0] =~ $re ) {
         $ignored{$re_name}++;
         next LINE;
      }
   }

   ...
}
[download]

If you can live without %ignored,

use Regexp::Assemble qw( )

{
   my %patmap = (
      '*' => '.*',
      '?' => '.',
      '[' => '[',
      ']' => ']',
   );

   my ($norm) = map qr/[^$_]/, join '', keys %patmap;

   sub glob2pat {
      my $globstr = @_ ? $_[0] : $_;
      $globstr =~ s{($norm+|.)} { $patmap{$1} || "\Q$1" }sge;
      return "^$globstr$";
   }
}

my $ignore_re = do {
   my $ra = Regexp::Assemble->new();
   $ra->add(glob2pat($_)) for @list_regexps;
   $ra->re()
};

while(<FILE>) {
   chomp;
   my( @cols ) = split(" ", $_);

   ...

   next if $cols[0] =~ $ignore_re;

   ...
}
[download]