There were many discussions about MathJax in chat both on this meta and on per-site-metas. (One of the main arguments why it might be useful seems to be that comments are rendered differently when they are displayed on the site and after they are moved to chat.)
Shog9's answer Any chance of MathJax in chat? says that:
This request was discussed by the team on April 9th, 2014. The potential rendering cost would be such that it would need to be a per-room and per-user option; this would then create a situation where users would potentially never be sure how their messages would look to other readers - in other words, little better than the user-script solution currently available.
Would somebody be willing to explain potential problems with MathJax in chat in more details? (And, if possible, in a way more accessible to a layman?)
Question. What are the problems that would be caused by enabling MathJax in chat? Would they be mostly on server side or client side? Would this mean significant slowing down of chat rendering for users? Would this be problem more for chat users on mobile devices than on desktop? (Although for mobile users there are some restrictions on viewing MathJax also on the main site.)
My motivation for asking this that in per-site-meta discussions, users often discuss whether or not the assessment of "potential rendering cost" is correct and what exactly are actual drawbacks from allowing MathJax in chat. (I can mention some discussions on meta's which I frequent: Why is there no support for MathJax in chat? or What's the point of having an MO chat without LaTeX enabled?) However, since the Stack Exchange team knows the details about the chat implementation, an answer from SE explaining the reasons behind this decision is likely to be more informative than comments by regular users - which could be, to some extent, considered mostly speculation.