I think this is fixed already.

After few hrs of debugging I found out why tex4ht was ignoring the font size on a \verb.

It turns out that adding \usepackage[T1]{fontenc} causes this.

Posted question on this here

http://tex.stackexchange.com/questions/318232/why-tex4ht-do-not-change-font-size-of-verb-when-including-usepackaget1font/318248#318248

MWE

\documentclass[11pt]{article}
\usepackage{ifluatex}
\ifluatex
\IfFileExists{luatex85.sty}{\usepackage{luatex85}}{}
\ifdefined\HCode
\usepackage[utf8]{luainputenc}
\usepackage[T1]{fontenc}
\else
\usepackage{fontspec}
\fi
\else%assume pdflatex, I only use these two
\usepackage[utf8]{inputenc}
\usepackage[T1]{fontenc}
\fi

\begin{document}
\ifluatex
compiled with lualatex
\else
compiled with pdflatex
\fi

{\tiny \verb|IllustratingTheUseOfDiscreteDistributions|}

{\tiny IllustratingTheUseOfDiscreteDistributions}
\end{document}

Compiling using make4ht foo.tex and looking at the HTML shows that the font size of the \verb did not change.

Removing \usepackage[T1]{fontenc} now the font size is correct.

TL 2016