How can I have multiple lines in a header or footer using scrpage2?

in LaTeX2e the command \footheight does not longer exist. It actually did in LaTeX 2.09. To solve your problem there are two possible options.

1) You can use \raisebox and e.g. a tabular environment to get everything below the line

\documentclass{scrartcl}

\usepackage{ucs}
\usepackage[utf8x]{inputenc}
\usepackage{layout}

\usepackage[headsepline, footsepline]{scrpage2}

\refoot{Course year\\ 2010/2011} 
\rofoot{\raisebox{-5ex}{\begin{tabular}[t]{rr}
Course year\\
2010/2011
\end{tabular}}}

\thispagestyle{scrheadings}
\pagestyle{scrheadings}

\begin{document}

 This is a sample document.
\end{document}

2) or you draw the line yourself with \rule

\documentclass{scrartcl}

\usepackage{ucs}
\usepackage[utf8x]{inputenc}
\usepackage{layout}

\usepackage[headsepline]{scrpage2}

\refoot{Course year\\ 2010/2011} 
\rofoot{%
\rule{1\textwidth}{1.5pt}
\begin{tabular}[t]{rr}
Course year\\
2010/2011
\end{tabular}}

\thispagestyle{scrheadings}
\pagestyle{scrheadings}

\begin{document}

 This is a sample document.
\end{document}

Since KOMA-Script version 3.12 there is a footheight option defined. Package scrlayer-scrpage (the successor of scrpage2) takes this option into account:

\documentclass[
  footheight=30pt
]{scrartcl}
\usepackage[utf8]{inputenc}

\usepackage[headsepline,footsepline]{scrlayer-scrpage}
\ofoot{Course year\\ 2010/2011}

\begin{document}
 This is a sample document.
\end{document}