marvosym symbols not working

The symbols can't be used in pure math mode. Remove the $...$ around it or wrap \text{} around the symbol macro name (\usepackage{amsmath}) then.

\documentclass{article}
\usepackage[english]{babel}
\usepackage{marvosym}

\begin{document}

\Bat
\Bicycle
\Coffeecup 
\Frowny 
\Heart
\Industry 
\Football
\Mobilefone
\Smiley
\Womanface
\Gentsroom
\Ladiesroom
\Taurus
\Sagittarius
\WomanFace
\ManFace
\Yingyang
\end{document} 

The symbol commands of marvosym are text mode commands, as explained in Christian Hupfer's answer.

Since marvosym's symbol macros are defined based on the more lower level macro \mvchr, these macros can be easily patched to get them work in math mode, e.g.:

\documentclass{article}
\usepackage[english]{babel}
\usepackage{marvosym}

\usepackage{amstext}
\usepackage{etoolbox}
\pretocmd\mvchr{\text}{}{\errmessage{Patching \noexpand\mvchr failed}}
\pretocmd\textmvs{\text}{}{\errmessage{Patching \noexpand\textmvs failed}}

\begin{document}

$\Bat$
$\Bicycle$
$\Coffeecup$
$\Frowny$
$\Heart$
$\Industry$
$\Football$
$\Mobilefone$
$\Smiley$
$\Womanface$
$\Gentsroom$
$\Ladiesroom$
$\Taurus$
$\Sagittarius$
$\WomanFace$
$\ManFace$
$\Yingyang$
\end{document}