Perhaps something like the following?

First, I mocked up your input file as you didn't give an example:

= a.txt
= c.txt
= b.txt
= e.txt
[download]

Here's the example code. Note I've changed your directory handle to a lexical one as opposed to a bareword global. Comments inline:

use warnings;
use strict;

my $directory_path = '.';

opendir my $dir, $directory_path or die "Cannot open directory $direct
+ory_path";

# create a hash with the files in the dir
# (makes lookup faster later on)

my %directory_path_file_list = map { $_ => 1 } readdir $dir;
closedir $dir;

my $text_file = "text_file_with_names.txt";

open (my $fh, '<', $text_file)
     or die("Can't open input file \"$text_file\": $!\n");

# define a new array for the files listed in the file, 
# extract the names, and push each filename onto the new
# array

my @files_in_file;

while (my $line = <$fh>) {
    if ($line =~ /=\s+(\w+\.txt)/){
        push @files_in_file, $1;
    }
}

# loop over the files found in the file, then check to see
# if each one exists in the hash we created earlier.
# you can perform your actions inside of the 'if' statement

for (@files_in_file){
    if (defined $directory_path_file_list{$_}){
        print "$_ file exists on disk\n";
    }
    else {
        print "$_ file doesn't seem to exist\n";
    }
}
[download]

Update: It slipped my mind to use -e as Eily suggested, but that's the way I'd go as well.