I've had nothing but trouble with Firefox on my older machines (a 2001-era 1.7GHz P4 w/ 256MB RAM that was retired as a regular desktop several years ago; a POS Dell with AMD64 dual-core and 2GB RAM).
On the other hand... I have to say, since getting my new laptop, at least on the Linux side... no memory issues whatsoever. It has been upgraded from 4GB to 16GB and the couple megabytes it used to swap to disk on occasion after long periods of use is no more, and the Windows side (although buggy as all hell), at least when it runs, actually runs nicely. The only trouble with Windows 8.1 is trying to get the damn thing to boot up without locking up. Oh, and the borked Start screen, which for some reason got all fucked up in one of the updates. Currently using just over 800MB RAM; i3 window manager on openSUSE 13.1, two xterm windows (one running wget), Geany text editor with a bunch of tabs, Dolphin file manager, and Firefox with five tabs. Seems to hover around 700-800MB regularly, usually only breaks a gig when I start a virtual machine.
To be fair, with this being a laptop instead of a desktop, I do tend to use it differently (though I almost always have it hooked up to my external monitor/mouse/keyboard and use it as a desktop). I shut it down every day and put it in its carrying case; the desktop was left on 24/7 and ready to go, set up as an SSH server for when I'm away. For this reason I often only have maybe a couple handfuls of tabs open in Firefox at any given time, instead of the dozens or hundred-plus that I used to have open on the old desktop. It should be interesting when I get a proper, modern desktop machine to replace that old one and use it in a way that is more "typical" to my previous usage patterns, but until then I'm stuck with what I have. Still, recall instant and horrendous swapping just by visiting gmail.com, no other tabs open, which is gone on this system; it might just bring it to close to a gig of RAM used.
I admit... I miss my always-on, always-ready, always-serving machine... the portability of the laptop is not quite making up for it. If I had to live with only one for the long term... desktop it is. Yet the laptop has its advantages, which I wouldn't want to do without either. I honestly don't see how people can get by with only a laptop, yet I hear people say it all the time.