I assume you are on Mac OS X and using Apple shipped version of Zapfino, then the answer is no; this is an AAT font and LuaLaTeX does not support AAT features. There is however an OpenType version of Zapfino by Linotype, it should work with LuaLaTeX (IIRC, Hans was using it as a benchmark while developing the OpenType code).
Update: here is an modified version of the aforementioned example to work with the OpenType font.
%!TEX TS-program = lualatex
%!TEX encoding = UTF-8 Unicode
% Dario Taraborelli (2008), Khaled Hosny (2011)
% The Beauty of LaTeX
% URL: http://nitens.org/taraborelli/latex
% Some rights reserved: CC-BY-SA
\documentclass[11pt]{article}
\usepackage{xcolor, fontspec, graphicx}
\definecolor{lightblue}{rgb}{.2,.5,1}
\newcommand{\red}[1]{\color{red} #1}
\newcommand{\reda}[1]{\fontspec[Color=red,Variant=2]{Zapfino Extra LT Pro}#1}
\newcommand{\redb}[1]{\fontspec[Color=red,Variant=3]{Zapfino Extra LT Pro}#1}
\newcommand{\redc}[1]{\fontspec[Color=red,Variant=4]{Zapfino Extra LT Pro}#1}
\newcommand{\redd}[1]{\fontspec[Color=red,Variant=5]{Zapfino Extra LT Pro}#1}
\newcommand{\rede}[1]{\fontspec[Color=red,Variant=6]{Zapfino Extra LT Pro}#1}
\newcommand{\redf}[1]{\fontspec[Color=red,Variant=7]{Zapfino Extra LT Pro}#1}
\newcommand{\redg}[1]{\fontspec[Color=red,Variant=8]{Zapfino Extra LT Pro}#1}
\newcommand{\lbl}[1]{\color{lightblue} #1}
\newcommand{\lbla}[1]{\fontspec[Color=lightblue,Variant=2]{Zapfino Extra LT Pro}#1}
\newcommand{\lblb}[1]{\fontspec[Color=lightblue,Variant=3]{Zapfino Extra LT Pro}#1}
\newcommand{\lblc}[1]{\fontspec[Color=lightblue,Variant=4]{Zapfino Extra LT Pro}#1}
\newcommand{\lbld}[1]{\fontspec[Color=lightblue,Variant=5]{Zapfino Extra LT Pro}#1}
\newcommand{\lble}[1]{\fontspec[Color=lightblue,Variant=6]{Zapfino Extra LT Pro}#1}
\newcommand{\lblf}[1]{\fontspec[Color=lightblue,Variant=7]{Zapfino Extra LT Pro}#1}
\newcommand{\lblg}[1]{\fontspec[Color=lightblue,Variant=8]{Zapfino Extra LT Pro}#1}
\newcommand{\old}[1]{%
\fontspec[Ligatures={Common, Rare}]{Zapfino Extra LT Pro}%
\fontsize{25pt}{30pt}\selectfont #1}
\begin{document}
\thispagestyle{empty}
\old
\noindent{\red d}roo{\lbl g}
\noindent{\reda d}roo{\lbla g}
\noindent{\redb d}roo{\lblb g}
\noindent{\redf d}roo{\lblf g}\\[.3cm]
\noindent{\redc d}roo{\lblc g}
\noindent{\redd d}roo{\lbld g}
\noindent{\rede d}roo{\lble g}
\noindent{\redg d}roo{\lblg g}\\[.2cm]
\end{document}
PS: This can be written in a more compact way, but I didn't want to diverge that much from the original example.
