Set vertical spacing between two particular items

You can use \smash{\overset\circ K_{n+1}} to hide the height of that element.

\documentclass[a4paper]{article}
\usepackage{amsmath,amssymb}
\newtheorem{theorem}{Theorem}[section]
\begin{document}

\begin{theorem}
There is a $(K_n)_{n\in\mathbb N}\subseteq E$ with
\begin{itemize}
    \item $K_n$ is compact;
    \item $K_n\subseteq\smash{\overset\circ K_{n+1}}$; and
    \item $d(\partial K_n,\partial K_{n+1})>0$
\end{itemize}
for all $n\in\mathbb N$.
\end{theorem}

\end{document} 

You don't have this problem if you use the \mathring accent:

\documentclass[a4paper]{article}
\usepackage{amsmath,amssymb}
\newtheorem{theorem}{Theorem}[section]

\begin{document}

\begin{theorem}
There is a $(K_n)_{n\in\mathbb N}\subseteq E$ with
\begin{itemize}
    \item $K_n$ is compact;
    \item $K_n\subseteq \ring{K}_{n+1} $; and
    \item $d(\partial K_n,\partial K_{n+1})>0$
\end{itemize}
for all $n\in\mathbb N$.
\end{theorem}

\end{document}