Commutative diagram using Tikzcd in footnote

Welcome to TeX-SE! You need to use an ampersand replacement (or another trick), but then it works.

\documentclass{article}
\usepackage{tikz-cd}
\begin{document}
\[\begin{tikzcd}A&B\\C&D\end{tikzcd}\] 

X\footnote{\[\begin{tikzcd}[ampersand replacement=\&]A\&B\\C\&D\end{tikzcd}\]}
\end{document}