Use the optional argument of \caption:
\caption[<Figure title>]{<Figure title> \cite{<Source>}}
EDIT: You'd "still like the citation to appear in the caption in the listoffigures but just to have the number as if it were from it's position in the main body of text". I don't know how to do this using the ieeetr style, but it works out of the box using biblatex:
\documentclass{article}
\usepackage[sorting=none]{biblatex}
\usepackage{filecontents}
\begin{filecontents}{\jobname.bib}
@misc{A01,
author = {Author, A.},
year = {2001},
title = {Alpha},
}
@misc{B02,
author = {Buthor, B.},
year = {2002},
title = {Bravo},
}
\end{filecontents}
\addbibresource{\jobname.bib}
\begin{document}
\listoffigures
\section{foo}
Some text \autocite{B02}.
\begin{figure}[!h]
\centering
\rule{1cm}{1cm}
\caption{A figure \autocite{A01}}
\end{figure}
\printbibliography
\end{document}

\usepackage{notoccite}in the preamble does the magic. – Jun 09 '15 at 23:25