A simple way to do this is to use \centerline{..} around the minipage. (This does not allow for verbatim or similar special content, but there is a \Centerline variant from the realboxes package, which does!)
I would add some small or medium vertical skip before and after this minipage to get decent vertical spacing.
\documentclass{article}
\usepackage{lipsum}% dummy text
\begin{document}
\lipsum[1]
\par\smallskip\noindent
\centerline{\begin{minipage}{1.5\textwidth}
\lipsum[2]
\end{minipage}}
\par\smallskip
\lipsum[3]
\end{document}
Another easy way is to use the adjustbox environment from the package with the same name. It provides the keys minipage=<minipage args>, center(=<length>, normally \linewidth) and also margin(=<left/right> <top/bottom>, and more).
\documentclass{article}
\usepackage{adjustbox}
\usepackage{lipsum}% dummy text
\begin{document}
\lipsum[1]
\begin{adjustbox}{minipage=1.5\textwidth,margin=0pt \smallskipamount,center}
\lipsum[2]
\end{adjustbox}
\lipsum[3]
\end{document}
Compare the similar question and answers at How can I center a too wide table?.