Adding two latex variables in Tikz beamer

Just use \the\numexpr\i+\j. This works regardless of whether or not you have loaded TikZ. (And use math mode and \scriptstyle instead of \tiny.)

\documentclass{beamer}
\usepackage{tikz}
\usetikzlibrary{matrix}

\setbeamertemplate{footline}[frame number]{}
\setbeamertemplate{navigation symbols}{}
\setbeamertemplate{footline}{}
\setbeamertemplate{frametitle}[default][center]

\begin{document}

\title{Some Title}
\author{greatest author out there}
\frame{\titlepage}
\begin{frame}
\frametitle{Adding two integers}
\begin{tikzpicture}
\def \i {4}
\def \j {17}
\draw (0,0) node{$\scriptstyle i+j =\the\numexpr\i+\j$};
\end{tikzpicture}
\end{frame}
\end{document}

Try this way:

\documentclass{beamer}
\usepackage{tikz}
\usetikzlibrary{matrix}

\setbeamertemplate{footline}[frame number]{}
\setbeamertemplate{navigation symbols}{}
\setbeamertemplate{footline}{}
\setbeamertemplate{frametitle}[default][center]

\begin{document}

\title{Some Title}
\author{greatest author out there}
\frame{\titlepage}
\begin{frame}
\frametitle{Adding two integers}
\begin{tikzpicture}
\def \i {4}
\def \j {17}
\pgfmathtruncatemacro\ij{\i+\j}
\draw (0,0) node{\tiny i+j = \ij};
\end{tikzpicture}
\end{frame}

\end{document}

You have to tell tikz/pgf to calculate the result first ... This is what \pgfmathtruncatemacro does here.

PS: See also \pgfmathsetmacro

tikz does not know that you want the sum, you need to tell it with \pgfmathparse

\tiny{...} is wrong. It should be {\tiny ...}, but better use the tikz key font=\tiny

\documentclass{beamer}
\usepackage{tikz}
\usetikzlibrary{matrix}

\setbeamertemplate{footline}[frame number]{}
\setbeamertemplate{navigation symbols}{}
\setbeamertemplate{footline}{}
\setbeamertemplate{frametitle}[default][center]

\begin{document}

\title{Some Title}
\author{greatest author out there}
\frame{\titlepage}
\begin{frame}
\frametitle{Adding two integers}
\begin{tikzpicture}
\def \i {4}
\def \j {17}

\draw (0,0) node[font=\tiny] {i+j =\pgfmathparse{\i+\j}\pgfmathresult };
\end{tikzpicture}
\end{frame}

\end{document}