Here's a solution that works with BibTeX and the natbib citation management package. It's admittedly quite kludgy. If someone's got a better solution, please step forward.
The solution consists of three parts. First, add the following lines to the top of the .bib file:
@preamble{ " \newcommand{\noopsort}[1]{} "
# " \newcommand{\bibstar}{\textsuperscript{*}} " }
If necessary, modify the definition of \bibstar so that its output conforms to your needs. The macro \noopsort seems to do nothing but, as will become clear soon, is very useful for sorting purposes.
Second, for each entry that's supposed to be prefixed with a "star" in the references section, modify the (first) author's name as follows: Instead of, say,
author = {Henry Jones},
write
author = {Henry \noopsort{Jones}\bibstar{}Jones},
The instruction \noopsort{Jones} ensures that the entry will be sorted under "J" for "Jones" rather than under whatever BibTeX decides the sorting order may be for "*" (probably after "Z").
Third, for each starred entry, define a "citation alias" using the mechanism provided by natbib. E.g., if the entry for Henry Jones's piece has the key jones11, you'd write
\defcitealias{jones11}{Jones (2011)}
in the preamble (after the natbib package has been loaded).
With these three things put in place, use \citetalias rather than \citet to cite the entry in question, i.e., write \citetalias{jones11}. (If you use \citet you'd get, surprise, *Jones (2011)...)
A full example:

\RequirePackage{filecontents}
\begin{filecontents*}{\jobname.bib}
@preamble{ " \newcommand{\noop}[1]{} "
# " \newcommand{\bibstar}{\textsuperscript{*}} " }
@article{jones11,
author={Henry \noop{Jones}\bibstar{}Jones},
journal={Circularity Today},
title={Deep thoughts},
year={2011},
volume = 1,
issue = 1,
pages = "1-101",
}
\end{filecontents*}
\documentclass{article}
\usepackage[authoryear]{natbib}
\bibliographystyle{apalike}
\defcitealias{jones11}{Jones (2011)}
\begin{document}
\citetalias{jones11}
\bibliography{\jobname}
\end{document}