Tufte-handout caption overlapping figure

You can wrap your tabular environment in a command that makes a box, such as makebox:

    \makebox[\textwidth]{%
        \begin{tabular}{ccc}
            \emptyfig & \emptyfig & \emptyfig \\
            Hi        & Hi        & Hi
        \end{tabular}%
    }

% arara: pdflatex
% !arara: indent: {overwrite: yes}
\documentclass[nohyper,nofonts]{tufte-handout}
\usepackage{graphicx}
\usepackage{lipsum}
\newcommand{\emptyfig}{\includegraphics[width=.3\textwidth]{example-image-a}}
\begin{document}
\begin{figure*}
    \makebox[\textwidth]{%
        \begin{tabular}{ccc}
            \emptyfig & \emptyfig & \emptyfig \\
            Hi        & Hi        & Hi
        \end{tabular}%
    }
    \caption{This is an overlapping caption}
\end{figure*}
\end{document}

For what it's worth, there are other solutions:

Option 1

Simply add \vspace*{desired space} before the caption:

\documentclass[nohyper,nofonts]{tufte-handout}
\usepackage{pgfplots}
\usepackage{lipsum}
\newcommand{\emptyfig}{\begin{tikzpicture}%
    \begin{axis}[width=\textwidth/3,height=\textwidth/3]%
            %\addplot[black] coordinates { (0,0) (1,1) };
    \end{axis}%
\end{tikzpicture}}
\begin{document}
\begin{figure*}
    \begin{tabular}{ccc}
            \emptyfig & \emptyfig & \emptyfig \\
            Hi & Hi & Hi
    \end{tabular}
    \vspace*{1cm}
    \caption{This is an overlapping caption}
\end{figure*}
\end{document}

Option 2

Use optional parameter for the \caption command:

\documentclass[nohyper,nofonts]{tufte-handout}
\usepackage{pgfplots}
\usepackage{lipsum}
\newcommand{\emptyfig}{\begin{tikzpicture}%
    \begin{axis}[width=\textwidth/3,height=\textwidth/3]%
            %\addplot[black] coordinates { (0,0) (1,1) };
    \end{axis}%
\end{tikzpicture}}
\begin{document}
\begin{figure*}
    \begin{tabular}{ccc}
            \emptyfig & \emptyfig & \emptyfig \\
            Hi & Hi & Hi
    \end{tabular}
    \caption[][6mm]{This is an overlapping caption}
\end{figure*}
\end{document}