This begs for the use of an 'ini' file and to exploit the lookup abilities of a hash.

As it appears that the url is embedded in other stuff, and you appear to want to root pathed urls to the root of the redirects--Is that what the \W => \n stuff is doing?--then you can't use a straight lookup in a hash.

You first need to separate out the domain name for the lookup, and substitute that back into the original string.

#! perl -slw
use strict;
use Data::Dumper;

my %map = map{ split /=/, $_, 2 } split /\n/, do{ local $/; <DATA> };

print Dumper \%map;

while( <> ) {
    chomp;
    if( m[http://([^/]+)/(\W?)]i ) {
        next unless defined $map{ $1 };
        substr( $_, $-[0], $+[0] -  $-[0] ) =
            'http://' . $map{ $1 } . ( length $2 ? $2 : "\n$2" );
    }
    print;
}

__DATA__
www.theirsite.com=our.server1/theirsite/
www.dummy.com=our.server2/dummy/
[download]

P:\test>type junk.txt
http://www.theirsite.com/
http://www.dummy.com/
http://www.theirsite.com/stuff
http://www.dummy.com/stuff
http://www.theirsite.com/ stuff
http://www.dummy.com/ stuff

P:\test>perl 324406-2.pl < junk.txt
$VAR1 = {
          'www.theirsite.com' => 'our.server1/theirsite/',
          'www.dummy.com' => 'our.server2/dummy/'
        };

http://our.server1/theirsite/
http://our.server2/dummy/
http://our.server1/theirsite/
stuff
http://our.server2/dummy/
stuff
http://our.server1/theirsite/ stuff
http://our.server2/dummy/ stuff
[download]

You could probably use a better regex for getting the domain and one of the Config::* modules for parsing the ini file.