\documentclass{article}
\input tex4ht.sty
\Preamble{html,3,next,fonts}
\begin{document}
\EndPreamble
The mathit needs a, preferable built-in, Protect.
\section{What is $\mathit{mathit}$?}
section one.
$\mathit{NONE-mathit}$\par
$\mathrm{NONE-mathrm}$\par
$\mathsf{NONE-mathsf}$\par
$\mathbf{NONE-mathbf}$\par
$\mathtt{NONE-mathtt}$\par
\par
\textbf{NONE-textbf}\par
\textit{NONE-textit}\par
\textrm{NONE-textrm}\par
\textsc{NONE-textsc}\par
\textsf{NONE-textsf}\par
\textsl{NONE-textsl}\par
\texttt{NONE-texttt}\par
\section{What is \textit{textit}?}
test
\end{document}