Remove the colored section box from moderncv

Write after your given two lines to change the color of the leading box to white:

\definecolor{mycolor}{rgb}{255,255,255}   % color white
\makeatletter                             % make @ a letter
\renewcommand*{\section}[1]{%
  \par\addvspace{2.5ex}%
  \phantomsection{}% reset the anchor for hyperrefs
  \addcontentsline{toc}{section}{#1}%
  \parbox[t]{\hintscolumnwidth}{\strut\raggedleft\raisebox{\baseletterheight}{\color{mycolor}\rule{\hintscolumnwidth}{0.95ex}}}%
  \hspace{\separatorcolumnwidth}%
  \parbox[t]{\maincolumnwidth}{\strut\sectionstyle{#1}}%
  \par\nobreak\addvspace{1ex}\@afterheading}% to avoid a pagebreak after the heading
\makeatother

This workaround runs with the styles casual and classic but not with banking. I changed only the used color from \color{color1} to \color{mycolor} and set mycolor to white. The original definition of macro section you can find in the file moderncvstyleclassic.sty.

Have you checked the style banking? This style does not use colored boxes.

Update to class moderncv, version 2.0.0 (current version):

with the current class moderncv, version 2.0.0 the definition in style casual has changed to:

\RenewDocumentCommand{\section}{sm}{%
  \par\addvspace{2.5ex}%
  \phantomsection{}% reset the anchor for hyperrefs
  \addcontentsline{toc}{section}{#2}%
  \cvitem[0ex]{\strut\raggedleft\raisebox{\baseletterheight}{\color{color1}\rule{\hintscolumnwidth}{0.95ex}}}{\strut\sectionstyle{#2}}%
  \par\nobreak\addvspace{1ex}\@afterheading}% to avoid a pagebreak after the heading

Therefore we need to use command \RenewDocumentCommand to redefine \color{color1} to \color{white}. Please note that a patch with command \patchcmd is not working, commands defined with \RenewDocumentCommand are not patchable!

We need to add the following code into the preamble (\makeatletter and \makeatother are needed because the code uses @):

\makeatletter
\RenewDocumentCommand{\section}{sm}{%
  \par\addvspace{2.5ex}%
  \phantomsection{}% reset the anchor for hyperrefs
  \addcontentsline{toc}{section}{#2}%
  \cvitem[0ex]{\strut\raggedleft\raisebox{\baseletterheight}{\color{white}\rule{\hintscolumnwidth}{0.95ex}}}{\strut\sectionstyle{#2}}%
  \par\nobreak\addvspace{1ex}\@afterheading}% to avoid a pagebreak after the heading
\makeatother

With the complete code

\documentclass[11pt,a4paper,sans]{moderncv}

\moderncvstyle{casual} % casual, classic, banking, oldstyle and fancy  % head 2, body 1, foot 1
\moderncvcolor{black} 

\usepackage[utf8]{inputenc}

\usepackage[scale=0.75]{geometry}

\makeatletter
\RenewDocumentCommand{\section}{sm}{%
  \par\addvspace{2.5ex}%
  \phantomsection{}% reset the anchor for hyperrefs
  \addcontentsline{toc}{section}{#2}%
  \cvitem[0ex]{\strut\raggedleft\raisebox{\baseletterheight}{\color{white}\rule{\hintscolumnwidth}{0.95ex}}}{\strut\sectionstyle{#2}}%
  \par\nobreak\addvspace{1ex}\@afterheading}% to avoid a pagebreak after the heading
\makeatother

% personal data
\name{John}{Doe}
\title{Resumé title}
\address{street and number}{postcode city}{country}
\phone[mobile]{+1~(234)~567~890}
\phone[fixed]{+2~(345)~678~901}
\phone[fax]{+3~(456)~789~012}
\email{[email protected]}
\homepage{www.johndoe.com}
\social[linkedin]{john.doe}
\social[twitter]{jdoe}
\social[github]{jdoe}
\extrainfo{additional information}
\photo[64pt][0.4pt]{example-image-a}
\quote{Some quote}

\setlength{\footskip}{70pt}


\begin{document}

\makecvtitle

\section{Education}
\cventry{year--year}{Degree}{Institution--3}{City--4}{\textit{Grade}--5}{Description--6}  % arguments 3 to 6 can be left empty
\cventry{year--year}{Degree}{Institution}{City}{\textit{Grade}}{Description}

\section{Master thesis}
\cvitem{title}{\emph{Title}}
\cvitem{supervisors}{Supervisors}
\cvitem{description}{Short thesis abstract}

\section{Experience}
\subsection{Vocational}
\cventry{year--year}{Job title}{Employer}{City}{}{General description 
  no longer than 1--2 lines.\newline{}%
Detailed achievements:%
\begin{itemize}%
\item Achievement 1;
\item Achievement 2, with sub-achievements:
  \begin{itemize}%
  \item Sub-achievement (a);
  \item Sub-achievement (b), with sub-sub-achievements (don't do this!);
    \begin{itemize}
    \item Sub-sub-achievement i;
    \item Sub-sub-achievement ii;
    \item Sub-sub-achievement iii;
    \end{itemize}
  \item Sub-achievement (c);
  \end{itemize}
\item Achievement 3.
\end{itemize}}
\cventry{year--year}{Job title}{Employer}{City}{}{Description 
  line 1\newline{}Description line 2}
\subsection{Miscellaneous}
\cventry{year--year}{Job title}{Employer}{City}{}{Description}

\end{document}

you get your wished result:

Adding

\usepackage{etoolbox}% http://ctan.org/pkg/etoolbox
\patchcmd{\section}% <cmd>
  {\strut}% <search>
  {\strut\renewcommand{\raisebox}[2]{}}% <replace>
  {}{}% <success><failure>

to your document preamble removes the boxes altogether by making the \raisebox command equivalent to a no-op, thereby gobbling its two arguments. This redefinition of \raisebox is local, since it is executing inside a \parbox.

The output below is obtained using the casual template (template.tex) with a black style and using the image example-image-16x10 (from the mwe package) for the \photo: