If you read the first list into a hash instead of an array, you can then use a hash slice with the second array to get the intersection.

#usr/bin/perl
use strict;

my $name_1 = q{list_1.txt};
my $name_2 = q{list_2.txt};

open my $INFILE_1, q{<}, $name_1 or die "Can't open $name_1 : $!";
open my $INFILE_2, q{<}, $name_2 or die "Can't open $name_2 : $!";

my %list_1;                            # changed
while (my $line_1 = <$INFILE_1>){
    $line_1 =~ s/\s+$//;
    $list_1{$line_1} = $line_1;        # changed
}

my @list_2;
while (my $line_2 = <$INFILE_2>){
    $line_2 =~ s/\s+$//;
    push @list_2, $line_2;
}

close $INFILE_1;
close $INFILE_2;

# changed from here down
my @intersection = grep { defined } @list_1{@list_2};

foreach my $line ( @intersection ) {
    print "$line\n";
}
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This also requires that list 2 not have duplicate elements.