Latex-Beamer: overprint-environment wobbles

Not very elegant perhaps, but you could simply add some negative vspace before the align*.

\documentclass[xcolor=dvipsnames]{beamer}
\mode<presentation>
\usetheme{Boadilla}

\usepackage[ngerman]{babel}
\usepackage[utf8]{inputenc}
\usepackage{amsfonts,amsmath,amssymb,amsthm}

\DeclareMathOperator{\Var}{Var}
\DeclareMathOperator{\E}{\mathbb E} 

\title{Dummy-Titel}
\subtitle{Dummy-Untertitel}
\author{Autor}

\date{\today}


\begin{document}

\begin{frame}{Erwartungswert}
    \begin{itemize}
        \item $\E(X_{t+1} | X_t = i) = \sum_{j=0}^{2N} j \cdot P(X_{t+1}=j | X_t = i)$
        \item $X_{t+1}$ binomialverteilt mit Parametern $n=2N$ und $p=\frac{i}{2N}$:
        $\Rightarrow \E(X_{t+1} | X_t=i) = n \cdot p = i$
        \item Martingal-Eigenschaft: $\E(X_{t+1} | X_t) = X_t$
    \end{itemize}
        \begin{overprint}
            \onslide<1>
            \vspace{-.5cm}
            \begin{align*}
                \E(X_t) &= \E(\E(X_{t+1} | X_t)) = \sum_{i=0}^{2N} E(X_{t+1} | X_t = i) \cdot P(X_t = i) \\
                &= \sum_{i=0}^{2N} \left( \sum_{j=0}^{2N} j \cdot P(X_{t+1}=j | X_t = i) \right) \cdot P(X_t = i) \\
                &= \sum_{j=0}^{2N} j \cdot \left( \sum_{i=0}^{2N} \cdot P(X_{t+1}=j , X_t = i) \right) \\
                &= \sum_{j=0}^{2N} j \cdot P(X_{t+1}=j) = \E(X_{t+1})
            \end{align*}
            \onslide<2>
            \begin{equation*}
                \Rightarrow \E(X_0) = \E(X_1) = \E(X_2) = \ldots
            \end{equation*}
        \end{overprint}
\end{frame}
\end{document}

Here is another alternative that provide no "wobble" in the transition from one slide to the next.

Instead of using the overprint environment, individual elements are made visible/not by using a combination of \only and \visible overlay specifications. For optimal vertical alignment between the two slides, \vphantom supplied the appropriate vertical spacing in the first line of align* (due to the summation).

\documentclass[xcolor=dvipsnames]{beamer}
\mode<presentation>
\usetheme{Boadilla}

\usepackage[ngerman]{babel}
\usepackage[utf8]{inputenc}
\usepackage{lmodern,amsfonts,amsmath,amssymb,amsthm}

\DeclareMathOperator{\Var}{Var}
\DeclareMathOperator{\E}{\mathbb E} 

\title{Dummy-Titel}
\subtitle{Dummy-Untertitel}
\author{Autor}

\date{\today}


\begin{document}

\begin{frame}{Erwartungswert}
  \begin{itemize}
    \item $\E(X_{t+1} | X_t = i) = \sum_{j=0}^{2N} j \cdot P(X_{t+1}=j | X_t = i)$
    \item $X_{t+1}$ binomialverteilt mit Parametern $n=2N$ und $p=\frac{i}{2N}$:
      $\Rightarrow \E(X_{t+1} | X_t=i) = n \cdot p = i$
    \item Martingal-Eigenschaft: $\E(X_{t+1} | X_t) = X_t$
  \end{itemize}%
  \begin{align*}
    \visible<1>{\E(X_t)} 
      &\only<1>{= \E(\E(X_{t+1} | X_t)) = \sum_{i=0}^{2N} E(X_{t+1} | X_t = i) \cdot P(X_t = i)}%
       \only<2>{\Rightarrow \E(X_0) = \E(X_1) = \E(X_2) = \cdots \vphantom{\sum_{i=0}^{2N}}} \\
      &\visible<1>{= \sum_{i=0}^{2N} \left( \sum_{j=0}^{2N} j \cdot P(X_{t+1}=j | X_t = i) \right) \cdot P(X_t = i)} \\
      &\visible<1>{= \sum_{j=0}^{2N} j \cdot \left( \sum_{i=0}^{2N} \cdot P(X_{t+1}=j , X_t = i) \right)} \\
      &\visible<1>{= \sum_{j=0}^{2N} j \cdot P(X_{t+1}=j) = \E(X_{t+1})}
  \end{align*}
\end{frame}
\end{document}

The overprint environment itself may have left some gap in addition to what the align* environment would have left above it, causing your items to be shifted down so low. Additionally, the lmodern fonts supported some fonts that were missing from your original MWE.