Traced into LWP::UserAgent, and finally figured out that the parameters I passed into the credentials() call are wrong. Here is the fixed code:

use LWP::UserAgent;
use HTTP::Request;

use strict;
use warnings;

my $ua = new LWP::UserAgent();
$ua->credentials("localhost:8080", "Tomcat Manager Application", "peip
+", "930612");
my $req;
if ($ARGV[0] eq "remove") {
    $req = new HTTP::Request("GET", "http://localhost:8080/manager/rem
+ove?path=/hello1");
} elsif ($ARGV[0] eq "install") {
    $req = new HTTP::Request("GET", "http://localhost:8080/manager/ins
+tall?path=/hello1&war=jar:file:/jwstutorial13/examples/web/hello1/dis
+t/hello1.war!/");
}
my $res = $ua->request($req);
print $res->content();
[download]

I will continue work on it, to add Tk and XML parts. But first... Hey, my turkey is done!