I've been using a setup of LyX with XeTeX on my laptop and it's been working just fine. However, I installed it on my desktop (with the same fonts, packages, everything) and I've noticed that whenever I try to use emphasis (\emph) on Hebrew text it just doesn't work.
The problem seems to be specifically with the font David CLM, which is the default roman font and works fine on my other computer. With any other font set as the default, the problem disappears. I tried reinstalling the font on my desktop, and the italics option works fine in MS Word and other places, but when I try to emphasize text in LyX (and the same with making it slanted and italics via the menu) the output is just normal text.
If it isn't clear from the first paragraph, the LaTeX preamble is the same as the one on my laptop (which does work correctly) but I attached it here just in case:
%%% Set Hebrew and English fonts
\setmonofont[Script=Hebrew]{Miriam Mono CLM}
\newfontfamily{\englishfont}[Ligatures=TeX]{TeX Gyre Pagella}
%\newfontfamily{\englishfont}[Ligatures=TeX]{Latin Modern Roman}
% Proper font size in math mode.
\setmathrm[Ligatures=TeX]{Latin Modern Math}
%%% Another way is to set just the Hebrew fonts:
% \newfontfamily\hebrewfont[Script=Hebrew]{David CLM}
% \newfontfamily\hebrewfonttt[Script=Hebrew]{Miriam Mono CLM}
% \newfontfamily\hebrewfontsf[Script=Hebrew]{Simple CLM}
% consider using \usepackage{iffont}
%%% Sets both header and footer
\fancyhf{} % sets to nothing
\renewcommand{\headrulewidth}{0pt}
\fancyfoot[C]{\textup{\thepage}}
\renewcommand*{\refname}{Bibliography}
%%% Some alternative styles
\ifcsname question\endcsname
\theoremstyle{definition}
\newtheorem{quesh}[thm]{שאלה}
\renewenvironment{question}{\begin{quesh}}{\end{quesh}}
\else
\fi
\ifcsname sol*\endcsname
\theoremstyle{remark}
\newtheorem*{emphsol*}{פתרון}
\renewenvironment{sol*}{\begin{emphsol*}}{\end{emphsol*}}
\else
\fi
\renewcommand{\labelenumi}{(\alph{enumi})}
\renewcommand{\labelenumii}{\arabic{enumii})}
tools -> reconfigure
David CLM
. It is quite unclear how you use the font.\newfontfamily\hebrewfont[Script=Hebrew]{David CLM}
in the preamble, which I tried but it still doesn't work.