One caption in listoffigures for a ContinuedFloat

Simply use an empty optional argument for \caption in the continuation (and the optional argument in the first \caption if you want to get rid of "(cont)" in the list of figures):

\documentclass{article}
\usepackage[demo]{graphicx}
\usepackage{subfig}
\begin{document}

\listoffigures

\begin{figure}[!ht]
\centering
\subfloat[][]{\includegraphics[height=1cm]{fig1}}
\\
\subfloat[][]{\includegraphics[height=1cm]{fig2}}
\caption[A result that requires two pages.]{A result that requires two pages. \emph{(cont.)}}
\end{figure}

\begin{figure}
\ContinuedFloat
\centering
\subfloat[][]{\includegraphics[height=1cm]{fig3}}
\\
\subfloat[][]{\includegraphics[height=1cm]{fig4}}
\caption[]{A result that requires two pages.}
\end{figure}
\end{document}

I changed some of the settings so that the example will fit in just one page.

I originally didn't see the requirement concerning the page numbering in the LoF; here's now a solution:

\documentclass{article}
\usepackage[demo]{graphicx}
\usepackage{caption}
\usepackage{subfig}

\def\myaddcontentsline#1#2#3{%
\addtocontents{#1}{\protect\contentsline{#2}{#3}{\thepage--\finalpage}}}

\gdef\finalatpage#1{\global\def\finalpage{#1}}
\finalatpage{0}
\makeatletter
\newcommand\MyCaption[1]{%
  \myaddcontentsline{lof}{figure}{\protect\numberline{\thefigure}#1}}
  \newcommand\savefinalpage{\makeatletter
  \write\@auxout{\protect\finalatpage{\arabic{page}}}
  \makeatother}
\makeatother

\begin{document}

\listoffigures

\clearpage

\begin{figure}
  \captionsetup{list=no}
  \centering
  \subfloat[][]{\includegraphics[height=1cm]{fig1}}\\
  \subfloat[][]{\includegraphics[height=1cm]{fig2}}
  \caption{A result that requires two pages. \emph{(cont.)}}
  \MyCaption{A result that requires two pages}
\end{figure}

\clearpage\mbox{}\clearpage

\begin{figure}
  \captionsetup{list=no}
  \ContinuedFloat
  \centering
  \subfloat[][]{\includegraphics[height=1cm]{fig3}}\\
  \subfloat[][]{\includegraphics[height=1cm]{fig4}}
  \caption{A result that requires two pages.}
  \savefinalpage
\end{figure}

\end{document}

Now another version with the new requirements; in particular, it requires loading hyperref:

\documentclass{article}
\usepackage[demo]{graphicx}
\usepackage{caption}
\usepackage{subfig}
\usepackage[linktocpage,colorlinks=true]{hyperref}

\makeatletter
\def\myaddcontentsline#1#2#3{%
\addtocontents{#1}{\protect\contentsline{#2}{#3}%
    {\protect\hyperlink{xyzs}{\thepage}--\protect\hyperlink{xyzf}{\finalpage}}{}}%
}%

\gdef\finalatpage#1{\gdef\finalpage{#1}}
\finalatpage{0}

\def\MyCaption#1{\hypertarget{xyzs}{}%
  \myaddcontentsline{lof}{figure}{\protect\numberline{\thefigure}#1}%
}

\def\savefinalpage{\hypertarget{xyzf}{}\makeatletter
  \write\@auxout{\protect\finalatpage{\arabic{page}}}
  \makeatother
}
\makeatother

\begin{document}

\listoffigures

\clearpage

\begin{figure}
  \captionsetup{list=no}
  \centering
  \subfloat[][]{\includegraphics[height=1cm]{fig1}}\\
  \subfloat[][]{\includegraphics[height=1cm]{fig2}}
  \caption{A result that requires two pages. \emph{(cont.)}}
  \MyCaption{A result that requires two pages}
\end{figure}

\clearpage\mbox{}\clearpage

\begin{figure}
  \captionsetup{list=no}
  \ContinuedFloat
  \centering
  \subfloat[][]{\includegraphics[height=1cm]{fig3}}\\
  \subfloat[][]{\includegraphics[height=1cm]{fig4}}
  \caption{A result that requires two pages.}
  \savefinalpage
\end{figure}

\end{document}

A new version that improves the anchoring of the hyperlink for the continuation:

\documentclass{article}
\usepackage[demo]{graphicx}
\usepackage{caption}
\usepackage{subfig}
\usepackage[linktocpage,colorlinks=true]{hyperref}

\makeatletter
\def\myaddcontentsline#1#2#3{%
\addtocontents{#1}{\protect\contentsline{#2}{\ignorespaces #3}%
    {\protect\hyperlink{xyzs}{\thepage}--\protect\hyperlink{xyzf}{\finalpage}}{}}%
}%

\gdef\finalatpage#1{\gdef\finalpage{#1}}
\finalatpage{0}

\def\MyCaption#1{\hypertarget{xyzs}{}%
  \myaddcontentsline{lof}{figure}{\protect\numberline{\thefigure}#1}%
}

\def\savefinalpage{\makeatletter
  \write\@auxout{\protect\finalatpage{\arabic{page}}}
  \makeatother
}

\def\MyCaptionE#1{\caption[#1]{\hypertarget{xyzf}{#1}}%
  \savefinalpage
}

\makeatother

\begin{document}

\listoffigures

\clearpage

\begin{figure}
  \captionsetup{list=no}
  \centering
  \subfloat[][]{\includegraphics[height=1cm]{fig1}}\\
  \subfloat[][]{\includegraphics[height=1cm]{fig2}}
  \caption{A result that requires two pages. \emph{(cont.)}}
  \MyCaption{A result that requires two pages}
\end{figure}

\clearpage\mbox{}\clearpage
\begin{figure}
  \caption{test}
\end{figure}
\begin{figure}[!b]
  \captionsetup{list=no}
  \ContinuedFloat
  \centering
  \subfloat[][]{\includegraphics[height=1cm]{fig3}}\\
  \subfloat[][]{\includegraphics[height=1cm]{fig4}}
  \MyCaptionE{A result that requires two pages.}
\end{figure}

\end{document}

Remark:

These last two codes deserve some explanation but unfortunately right now I won't be able to provide it. As soon as I can I will do so.