If you are on Windows and using a relatively old system, you may find that the K-Meleon browser offers that fastest experience in chess.com's live chess server. It doesn't pack the latest tech, but it still does a lot of things, it's very light on resources and very fast overall. It offers the best live chess experience on chess.com in my experience.
If you want to stick with Firefox, you may want to deactivate all the useless bloatware it comes with by tweaking the hidden "about:config" settings (type it in the URL address bar and press Enter). Careful with what you do there because you can easily make your browser perform much worse than stock if you don't know what you're doing. If you have extensively modified Firefox, it's more likely that any perceivable slowness or unresponsiveness derives from some of your tweaks.
I have highly customized Firefox installations running on Linux and Windows. Until a few days ago, I was also experiencing pretty bad 'lag' issues which made my bullet rating plummet 200 - 300 points below my average.
Just a few days ago (when I finally decided to look into it), I found out that the problem wasn't lag. The slowness was caused by some javascript resources that I had deactivated on my browser for security sake. The following settings:
"javascript.options.asmjs"
"javascript.options.ion"
"javascript.options.baselinejit"
It seems that chess.com's live chess interface relies heavily on javascript, so the above settings must be set to "true" (enabled) to enjoy the best live chess experience on Firefox. Obviously, javascript should be enabled in the browser - "javascript.enabled" must be "true" as well.
This affects not only the live chess interface, but also the engine analysis feature. The engine analysis on chess.com wasn't working right when I had the above settings set to "false."
So make sure that your browser settings and add-ons aren't jeopardizing javascript processes.
DOM objects also appear to be critical for the website's interface (v3). Any misconfiguration in your browser's DOM settings can render the website unresponsive.
Even if none of this applies, I still leave it here for whoever might be in the same situation. There are many other things that can be tweaked on Firefox to improve speed on chess.com's live chess, but some will depend on the host system.
I also have this problem, am on a mac and safari work fine but firefox has terrible lag, it is a new problem.