You often need to do something to a line depending on what is on the next line. One way to go is to store a line, move to the next line, decide what to do and then store the current line as appropriate.

#! /usr/perl/bin

use warnings;
use strict;

my $skip = 0;

my $last_line = <DATA>;
chomp($last_line);

while (my $current_line = <DATA>){
    chomp($current_line);
    if ($current_line =~ /^\d{4}-\d{2}-\d{2}/){
        if (not $skip){
            print $last_line, qq{\n};
        }
        else{
            $skip = 0;
        }
        
        $last_line = $current_line;
    }
    else{
        print $last_line, qq{ }, $current_line, qq{\n};
        $skip = 1;
    }
}

print $last_line, qq{\n};

__DATA__
2013-01-01 user1 user2 chat chat
2013-01-02 user3 user4 more chat more chat
2013-01-03 user5 user6 nazi nazi
chat chat nazi
2013-01-04 user 7 user8 see you in the pub
[download]

2013-01-01 user1 user2 chat chat
2013-01-02 user3 user4 more chat more chat
2013-01-03 user5 user6 nazi nazi chat chat nazi
2013-01-04 user 7 user8 see you in the pub
[download]

Boy, am I rusty. It hurt my head getting that to compile and run. :-)

Updated: Reset the $skip flag.

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