First, always use strictures like this:

#!/usr/bin/perl

use strict;
use warnings;
use Date::Manip;
[download]

Second, fix your lexical problem. For example,

$date_two_days_ago = ParseDate('3 days ago');
[download]

should be

my $date_two_days_ago = ParseDate('3 days ago');
[download]

Third, you used $arr[3] but forgot to give the array.
Fourth, run perltidy to cleanup. Here's what I got:

#!/usr/bin/perl

use strict;
use warnings;
use Date::Manip;

my $date_2_days_ago = ParseDate('3 days ago');
my $date_converted = UnixDate( $date_2_days_ago, '%e/%h/%Y' );

open DATA, '>', $ARGV[1];
open FH, $ARGV[0];

while (<FH>) {
    my(@tab_delimited_array) = split( /\t/, $_ );
    $tab_delimited_array[3] =~ s/^\[//;
    $tab_delimited_array[3] =~ s/^\-//;
    chomp $tab_delimited_array[3];
    if ( length $tab_delimited_array[3] > 1 ) {
        my $date_format = UnixDate( $tab_delimited_array[3], '%Y%m%d%H
+:%M:%S' );
        my $converted_date = Date_ConvTZ( $date_format, 'GMT', 'PST' )
+;
        my $pst_converted_date = UnixDate( $converted_date, '%e/%h/%Y:
+%H:%M:%S' );
        $pst_converted_date =~ s/^\s//g;
        my $extracted_YMD = UnixDate( $converted_date, '%e/%h/%Y' );
        my @arr;
        $_ =~ s/$arr[3]/$pst_converted_date/g;

        if ( $extracted_YMD =~ /$date_converted/ ) {
            print DATA $_;
        }
    }
}
close DATA;
close FH;
__DATA__
[download]

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