Suppressing whitespace at end of environment

Use \unskip before the ] in the closing part of the environment definition. This will eat up any space characters prior to the ].

EDIT to heed Barbara's comment to remove stray space following em-dash. And as she notes, one cannot have a blank line before \end{example} without automatically introducing a paragraph break.

\documentclass{article}
\newenvironment{example}{[\textit{Example} ---}{\unskip]}
\begin{document}
\begin{example}
This is an example.
\end{example}
\end{document}

Furthermore, if you wish not to have the blank space after the closing ], you need \ignorespacesafterend, as in

\documentclass{article}
\newenvironment{example}{[\textit{Example} ---}{\unskip]\ignorespacesafterend}
\begin{document}
\begin{example}
This is an example.
\end{example}
xxx
\end{document}

I'm not sure you really want an environment for this. However, you can (ab)use environ.

\documentclass{article}
\usepackage{environ}

\NewEnviron{example}{%
  [\textit{Example} --- \BODY]%
}

\begin{document}

\begin{example}
This is an example.
\end{example}

\end{document}