How to prevent page break between section heading and longtabu table?

Have a look at the needspace package. The functionality also exists in for example memoir. With it you can add something like: \needspace{5\baselineskip} before the headline. If not the given space is left on the page a page break will be inserted and in your case the header will end up on the next page.

From Absolutely, definitely, preventing page break:

Just enclose your text in an unbreakable unit; before it issue a combination of glue and penalties that will fill the page if the unit has to go to the next one. 

This works for your situation, and does not create awkward, unnecessary page breaks.

\documentclass{report}
\usepackage{longtable,tabu}

\newenvironment{absolutelynopagebreak}
  {\par\nobreak\vfil\penalty0\vfilneg
   \vtop\bgroup}
  {\par\xdef\tpd{\the\prevdepth}\egroup
   \prevdepth=\tpd}

\begin{document}
An initial line of text

\vskip 47.5em % Offset the section to close to the bottom of the page

\begin{absolutelynopagebreak}
\section{Test section}
\nopagebreak
\begin{longtabu}{l X}
    \textbf{Column 1} & \textbf{Column 2} \\
    \hline
    \\
    \endhead
    Cell 1 & Cell 2 \\
\end{longtabu}
\end{absolutelynopagebreak}

\end{document}