Fun exercise. ;-)
\documentclass{article}
\usepackage{regexpatch,fancyvrb,xparse}
\makeatletter
\let\do@footnotetext\@footnotetext
\regexpatchcmd{\do@footnotetext}
{\c{insert}\c{footins}\cB.(.*)\cE.}
{\1\c{egroup}}
{}{}
\def\@footnotetext{\insert\footins\bgroup\@makeother\#\do@footnotetext}
\newcommand{\ttvar}{\begingroup\@makeother\#\@ttvar}
\newcommand{\@ttvar}[1]{\ttfamily\detokenize{#1}\endgroup}
\makeatother
\setlength{\textheight}{3cm} % just to shorten the height for the example
\begin{document}
Works inline \ttvar{S#Var} and also in footnote.%
\footnote{Yes, \ttvar{S#Var}, as you see.}
\end{document}

With \detokenize you can print all characters, except backslashes and % (actually backslashes are allowed, but they might produce unwanted spaces), so long as braces are balanced.
The trick is to modify \@footnotetext not to absorb its argument, but just doing \insert\footins\bgroup, changing the category code of # (you quite certainly won't be defining commands in a footnote) and calling \do@footnotetext that will absorb the argument. Somebody might enjoy looking how \regexpatch does its work on a copy of the original \@footnotetext.
The command \ttvar will locally change the category code of #.
Alternative solution.
\documentclass{article}
\usepackage{fancyvrb,cprotect}
\begin{document}
Works inline \Verb!S#Var! and also in footnote.%
\cprotect\footnote{Yes, \Verb!S#Var!, as you see.}
\end{document}
\texttt(S\#Var}– David Carlisle Oct 19 '13 at 20:18\Verbdoesn't throw an error, but I believe the result will be unpredictable, in general. – egreg Oct 19 '13 at 20:19