Remove fancyhdr chapter 0 header

Change the \fancyhead[L]{...} to

\makeatletter
\fancyhead[L]{%
  % thank egreg for his help in comment
  \ifnum\value{chapter}=0\else\chaptername\ \thechapter\ --\ \fi\leftmark
}
\makeatother

and add

\chaptermark{Introduction}

after \chapter*{Introduction} help.

With this change,

• the second page of unnumbered chapter "Introduction" will have its left header showing "Introduction", and
• the second page of table of contents will have its left header showing "CONTENTS".

A full example

\documentclass{report}

\usepackage{fancyhdr}
\pagestyle{fancy}
\renewcommand{\chaptermark}[1]{\markboth{#1}{#1}}
\fancyhead[R]{}

\makeatletter
\fancyhead[L]{%
  % thank egreg for his help in comment
  \ifnum\value{chapter}=0\else\chaptername\ \thechapter\ --\ \fi\leftmark
}
\makeatother

\begin{document}

\tableofcontents

\chapter*{Introduction}
\chaptermark{Introduction}
\addcontentsline{toc}{chapter}{Introduction}
Text
\newpage
More text

\chapter{The first chapter}
\newpage
text

\end{document}

I'd use a different approach, namely to set the header in \chaptermark.

\documentclass{report}

\usepackage{fancyhdr}

\pagestyle{fancy}
\fancyhead{} % clear the headers
\fancyhead[L]{\nouppercase{\leftmark}}
\renewcommand{\chaptermark}[1]{\markboth{\chaptername\ \thechapter\ -- #1}{}}

\newcommand{\uchapter}[1]{%
  \chapter*{#1}%
  \markboth{#1}{}%
  \addcontentsline{toc}{chapter}{#1}%
}

\begin{document}

\tableofcontents

\uchapter{Introduction}

Text
\newpage
More text

\chapter{The first chapter}

Text
\newpage
More text

% check the header of a possible second page in the TOC
\addtocontents{toc}{\protect\newpage\protect\mbox{}}

\end{document}

The \nouppercase is needed because report adds \MakeUppercase to the arguments to \markboth for special chapters such as the TOC, the lists of figures and tables and the bibliography.

I introduced a \uchapter command to avoid heavy code in the document body.