What is involved in upgrading Android to a later version, if the manufacturer no longer supports the device? I recently acquired an Android-based tablet which is now running a rooted version 2.2; however, the manufacturer no longer supports or sells this particular tablet. 

I'm interested in seeing if I can install Android 4.0 onto it by keeping the Linux kernel from 2.2 and just replacing everything else. Despite the GPL, the manufacturer never released any source code for their custom modifications so I can't, for example, compile a new kernel. So my basic plan is to replace the software stack with the stock Android 4.0, copy any custom applications over from 2.2 that came with the device, and see what happens.

Has anyone here tried anything like this before and can advise on any potential pitfalls before I go ahead and inadvertently brick my device? Thanks in advance!
You might want to look for a "Custom ROM". For many devices, the community provides custom-built ROMs which bring newer Android versions to their devices -- often even improved over the manufacturers version (better optimized, no bloatware), and even covering newer versions where the manufacturer stopped.

A good source to find out if there are any custom ROMs available for your device and which is TheUnlockr, where everything is nicely grouped by manufacturer and device. Also, it seems to be the most complete source. Alternatively, there also is CommunityRelease (you can filter by device, Android version and developer) plus PDADB (no filters at all: over 800 pages to check, and not restricted to Android devices -- so it will be hard to find things here. Though, it may serve as a kind of "last ressort").

As you report your device to be rooted, the most important pre-condition to install a custom ROM is already met ;)