I was trying out the new option tex:dvipng. The HTML result was a image with the tikz code.
Turns out, debugging and telling the fragment processor not to delete the files, what org generates for the tikz fragment is:
\begin{tikzpicture*}
Removing the astherisk would make it work without any workarounds.
How I got around this was
#+LATEX_HEADER: \usepackage{environ,amsmath,multicol}
#+LATEX_HEADER: \NewEnviron{tikzpicture*}[1][]{\begin{tikzpicture}[#1]\BODY\end{tikzpicture}}
Possibly fixing this issue, will make cross formats much easier.