How to write the result of a citation to a file?

You can't use \cite because this has several instruction for printing the citation key.

You need to use \b@<key>, for instance like this:

\documentclass{article}

\newwrite\tempfile
\AtBeginDocument{\immediate\openout\tempfile=\jobname.txt}
\makeatletter
\newcommand{\writecite}[1]{%
  \@for\next:=#1\do{%
    \immediate\write\tempfile{%
      \@ifundefined{b@\next}{[<\next>]}{[\csname b@\next\endcsname]}%
    }%
  }%
}
\makeatother


\begin{document}
\cite{Ab,Cd}

\writecite{Ab,Cd}


\begin{thebibliography}{1}
\bibitem{Ab} Author, title, 2016.
\bibitem{Cd} Common, Second, 2015.
\end{thebibliography}

\end{document}

Contents after the first run

[<Ab>]
[<Cd>]

Contents after the second run

[1]
[2]