The documentation for mutt suggests using lynx with mailcap: http://www.mutt.org/doc/manual/#mimesupport

But using mailcap + some terminal browser should work in general. w3m, lynx, or browsh should work.

For example: http://jasonwryan.com/blog/2012/05/12/mutt/

Even a normal browser will work:

    text/html;      firefox -remote 'openURL(%s)'