Changing font size of list items and subitems

When working with beamer one always should use beamer syntax:

\documentclass{beamer}
\usetheme{Dresden}


\setbeamertemplate{itemize/enumerate body begin}{\large}
\setbeamertemplate{itemize/enumerate subbody begin}{\tiny}

\begin{document}

\begin{frame}
\frametitle{Frame title}

\begin{itemize}
\item First item 

\begin{itemize}
\item First subitem
\item Second subitem
\end{itemize}
\end{itemize}


\end{frame}

\end{document}

This simply sets the item body to a size. One level below (3rd) can be accessed via \setbeamertemplate{itemize/enumerate subsubbody begin}{<...>}. As one may encounter, the labels aren't properly aligned after the change of the fontsize. This is a result of beamers' manual like definition of the alignment of the labels. The orginal definition is \defbeamertemplate*{itemize item}{default}{\scriptsize\raise1.25pt\hbox{\donotcoloroutermaths$\blacktriangleright$}} so one needs to alter the size and the vertical position of labels. One may even need to alter the skips between the items.

Please always post complete documents that show the problem. I get consistent font size when I tried your fragment, but you may have conflicting packages.

In this case it seems to be a beamer feature which you can avoid by changing one line to say

\footnotesize \let\small\footnotesize

Makes \small locally the same as \foototesize so when beamer decides to insert \small no harm is done. (There may be a proper beamer configuration for this I don't know beamer that well)

Original answer:

I fixed a couple of markup errors (the inner {} are not needed and there should be a blank line before the closing } of a font size command. But they make no difference in this case.

\documentclass{article}

\begin{document}

{\footnotesize
\begin{itemize}
\item First item
\item Second item
\begin{itemize}
\item Third item
\item Fourth item
\end{itemize}
\end{itemize}

}

\end{document}