Unsupported operating systems
From MozillaZine Knowledge Base
Firefox 3.x and Thunderbird 3.x do not support Windows 98, Windows 98 SE, Windows ME, Windows NT 4.0, Mac OS X 10.2.x and Mac OS X 10.3.x. One workaround would be to use Firefox 184.108.40.206 and Thunderbird 220.127.116.11. However, it is very risky to use an obsolete browser because you can't get any security bugs for known vulnerabilities fixed, and exploits are out there. Using an obsolete version of Thunderbird is less (but still is) of a risk, there are less feasible exploits than for a browser. The major risks for e-mails are enabling inline attachments or blindly opening any attachment.
Windows 2000 and XP SP1 are no longer supported starting with Firefox 13 and Thunderbird 13 due to Mozilla moving to the Microsoft Visual C++ 2010 compiler. 
Firefox and Thunderbird do not officially support Windows 8 yet, but all reports are that it works fine. There are no plans to do a Metro version of Thunderbird. Since there is no Thunderbird "app" (Metro version of Thunderbird), Thunderbird won't run on a Surface RT tablet, since Windows RT only runs Windows Store apps. Thunderbird will run on the Surface Pro since it has the full Windows 8.     Mozilla is working on a Metro version of Firefox (its currently in beta). Mozilla would like to create a version of Firefox for Windows RT but Microsoft doesn't allow other software to use the APIs needed to support a modern web browser.   
If you have a unsupported version of Windows one workaround would be to install a kernel patch called kernelex. It designed to let applications that require Windows 2000 or Windows XP run on Windows 98/ME systems. It seems to have a good reputation but its recommended you assume the worst and back up your entire system and create a bootable rescue disk beforehand if you try it.
It adds a compatibility layer that emulates functions that the application expects to find. This includes several file and registry security features that it might pretend to implement, but actually ignore. Normally you would be more secure using Firefox 3.x since it has many security patches not in Firefox 18.104.22.168. However, its possible you could actually be less secure because Firefox 3 exposes you to some new and more serious vulnerabilities due to it assuming those features would protect you.
Another possibility is to create a browser and/or email appliance. You could run the latest version of Firefox and/or Thunderbird using a virtual machine that has a very lean copy of Linux (such as Damn Small Linux) as a guest operating system. This is not the same thing as dual booting. The problem is that most virtual machines (VMWare, VirtualBox etc.) are optimized to be run using a Windows 2000/XP host (rather than a Windows 98 host) so they're slow.
Mac OS X
There are various threads where somebody talks about somebody else finding a workaround, but never any details. SeaMonkey 1.1.19 will run on Mac OS X 10.2 or later. It was released for anybody unable to run the "newer stable version" on March 16, 2010 when support for 1.x was dropped. SeaMonkey will no longer backport security fixes to 1.x nightly builds.
Classilla is a open source browser for OS 9. Its built on WaMCom (Web and Mail Communicator), a port of Mozilla client software for classic Macintosh systems. Where possible it will also run on OS 8.6.
Mozilla plans on dropping support for Firefox on Leopard around June 2012. TenFourFour is a version of Firefox fine-tuned for PowerPC Macs running Tiger or Leopard. TenFourBird is a version of Thunderbird for PowerPC Mac OS X based on TenFourFox. As of April-5-2012 both TenFourFox and TenFourBird support the latest ESR (Extended Support Release) from Mozilla (10.0.3). The Camino browser runs on Tiger and later on both Intel and PowerPC Macs.