The default layout of the ToC for chapters (starred or not) under memoir
is to have no dots. This is set internally by
\newcommand{\cftchapterdotsep}{\cftnodots}
If you want dots, it's best to set it to match that of the sectional unit dots, which is the same as \cftdotsep
. As such, using
\renewcommand{\cftchapterdotsep}{\cftdotsep}% Chapters should use dots in ToC
works.

\documentclass{memoir}
\renewcommand{\cftchapterdotsep}{\cftdotsep}% Chapters should use dots in ToC
\begin{document}
\tableofcontents
\chapter{A chapter}
\chapter*{A starred chapter}
\addcontentsline{toc}{chapter}{\protect\numberline{}A starred chapter}
\section{A section}
\end{document}
\setcounter{secnumdepth}{-1}
instead of\chapter*
and\section*
all over. In this way you'll need no explicit\addcontentsline
. – egreg Oct 8 '14 at 8:17