There's no need to play with counters. If you have to "promote" only sections, the following will work:
\documentclass[article, a4paper, 12pt, oneside]{memoir}
\setlrmarginsandblock{23mm}{63mm}{*}
\setulmarginsandblock{23mm}{28mm}{*}
\setheadfoot{\onelineskip}{2\onelineskip}
\setheaderspaces{*}{1mm}{*}
% \chapterstyle{plain} % needed?
\checkandfixthelayout
\renewcommand{\thesection}{\arabic{section}}
\makeatletter
\let\l@section\l@chapter
\makeatother
\begin{document}
\tableofcontents
\section{Introduction} \label{beginning}
Beginning. Just wait til you see the end, that's in Chapter \ref{end}.
\section{Conclusion} \label{end}
End. Hopefully you read the beginning! (Chapter \ref{beginning})
\end{document}
If you have to promote also subsections, then
\renewcommand{\thesection}{\arabic{section}}
\renewcommand{\thesubsection}{\thesection.\arabic{subsection}}
\makeatletter
\let\l@subsection\l@section
\let\l@section\l@chapter
\makeatother
The strategy is to use for an entry in the table of contents the setting for the upper level (so we have to work from bottom up). The macro
\l@<sectional unit>
is responsible for typesetting the table of contents entry for that sectional unit.

Of course \tableofcontents*
should be used, in order to avoid that the table of contents is listed in itself, which makes little sense anyway.
Just use
{\let\clearpage\relax\chapter*{Test 2}}
for the second "chapter". Note that it's grouped, to localize the scope of the \clearpage
redefintion.

Best Answer
You could use the
article
option to simulate an article class. You don't need to call\chapter
, though it's still possible, having a\section
-like effect: similar heading, no page break. If this option has been set, numbering of figures, tables etc. is continuous.Example: