The MathJax web fonts do not include CJK characters, so if they are
used within your math (like in an <mtext> element or \text{} command),
MathJax will see them as unknown characters, and so will use a list of
fonts in hopes that one of them will contain the desired characters.
This list is stored in the "undefinedFamily" parameter of the HTML-CSS
and SVG sections of your MathJax configuration, so you can specify the
fonts you want to try. There is no guarantee that the viewer has
these fonts installed, however.
It is also possible to tell MathJax to render <mtext> and \text{}
content in the font currently in use on the page at the location where
the math is found rather than to try to use its own fonts. That is
controlled by the "mtextFontInherit" parameter in the HTML-CSS and SVG
sections of your configuration. If you set that to true then the
<mtext> and \text{} content should be visible in the same way as the
text in the rest of the page.
See
http://www.mathjax.org/docs/2.0/options/HTML-CSS.html
for more details on these parameters.
I don't know if there are any encoding issues to be worried about. If
you can use UTF-8 or UTF-16 encodings for the CJK characters on your
pages, then I think you should not have any trouble.
Davide