\renewcommand{\refname}{} does not remove "References"

If you are loading the package natbib, then

\renewcommand{\bibsection}{}

after \usepackage[<options>]{natbib} will get rid of the section automatically provided (and also of the blank space you'd get by only redefining \refname or \bibname).

For my part I used these command:

\bibliographystyle{plain} % bib style
    %%% change name of biliography
    \renewcommand{\bibname}{} %%no name

I just tested it on a article I wrote and it worked fine.