This is really weird, this seems to be a peculiarity of your setup. As most tests are passing, I would venture that it's safe to force-install, but it seems that at least sending cookies when downloading a file is broken.

I can't reproduce the problem locally, so maybe there is some finer stuff in the module (or software) versions. The output of this test file looks for me as follows:

#!perl -T

use strict;
use warnings;

use Test::More tests => 1;

my $module;

BEGIN {
   $module  = "WWW::Mechanize::Firefox";
   require_ok( $module );
}

diag( sprintf "Testing %s %s, Perl %s", $module, $module->VERSION, $] 
+);

for (sort grep /\.pm\z/, keys %INC) {
   s/\.pm\z//;
   s!/!::!g;
   eval { diag(join(' ', $_, $_->VERSION || '<unknown>')) };
}

__END__
 > perl -Ilib -wT t\00-load.t
1..1
ok 1 - require WWW::Mechanize::Firefox;
# Testing WWW::Mechanize::Firefox 0.59, Perl 5.012001
# AutoLoader 5.70
# CPAN::Meta::YAML 0.003
# Carp 1.16
# Config <unknown>
# Cwd 3.31
# DynaLoader 1.10
# Encode 2.39
# Encode::Alias 2.12
# Encode::Config 2.05
# Encode::Encoding 2.05
# Exporter 5.64_01
# Exporter::Heavy 5.64_01
# Fcntl 1.06
# File::Basename 2.78
# File::HomeDir::Driver 0.98
# File::Spec 3.31
# File::Spec::Unix 3.3
# File::Spec::Win32 3.3
# FileHandle 2.02
# Firefox::Application 0.59
# HTML::Selector::XPath 0.06
# HTTP::Cookies 6.00
# HTTP::Cookies::MozRepl 0.59
# HTTP::Cookies::Netscape 6.00
# HTTP::Date 6.00
# HTTP::Headers 6.00
# HTTP::Headers::Util 6.00
# HTTP::Message 6.02
# HTTP::Response 6.01
# HTTP::Status 6.00
# IO 1.25_02
# IO::File 1.14
# IO::Handle 1.28
# IO::Seekable 1.1
# JSON 2.53
# JSON::XS 2.3
# List::Util 1.23
# MIME::Base64 3.09
# MozRepl::RemoteObject 0.29
# MozRepl::RemoteObject::Methods 0.29
# Parse::CPAN::Meta 1.4401
# PerlIO 1.06
# Portable 0.13
# Portable::CPAN 0.13
# Portable::Config 0.13
# Portable::HomeDir 0.13
# Portable::minicpan 0.13
# Scalar::Util 1.23
# SelectSaver 1.02
# Storable 2.22
# Symbol 1.07
# Test::Builder 0.98
# Test::Builder::Module 0.98
# Test::More 0.98
# Time::HiRes 1.9721
# Time::Local 1.1901_01
# URI 1.54
# URI::Escape 3.30
# WWW::Mechanize::Firefox 0.59
# WWW::Mechanize::Link <unknown>
# XSLoader 0.10
# attributes 0.12
# base 2.15
# bytes 1.04
# common::sense 3.4
# constant 1.20
# overload 1.10
# parent 0.223
# re 0.11
# strict 1.04
# utf8 1.08
# vars 1.01
# warnings 1.09
# warnings::register 1.01
[download]

But I doubt that the difference really lies on the Perl side of things. Modifying t/50-app-new.t to also output the mozrepl version shows for my test setup that the tests pass across 1.1beta2 and 1.1.0:

    eval { @addons = $ff->addons; $lives++ };
    ok $lives, "We can query the addons"
        or diag $@;

    diag "Found " . scalar @addons . " addons";
    ok @addons >= 1, "You have at least one addon"; # The mozrepl addo
+n, duh

    my ($mozrepl) = grep { $_->{id} eq 'mozrepl@hyperstruct.net' } @ad
+dons;
    isn't $mozrepl, undef, "We find the mozrepl addon";
    is $mozrepl->{name}, 'MozRepl', 'The name is "MozRepl"';
    diag "Using MozRepl version $mozrepl->{version}";
[download]

If you have any other mozrepl version, that could be an easy solution, but you already said you use 1.1.0.