From the fan out on the question we see that IE7 appears to have been supported until sometime this year (presumably largely due to its prevalence on corporate desktops). Perhaps ESR versions of Firefox merit support for similar reasons.
The question is what the title says: Should Stack Exchange sites support ESR versions of Firefox?
As an example of why this might be desirable, I've been working as a contractor for a number of years and I've encountered more than one occasion where upgrading the browser was not an option, mainly due to incumbent 'no install' policies.
10.0.4 ESR is the vendor-supported browser on RHEL, so I could imagine a situation where policy requires the use of this browser. However, I don't know whether the there is a significant body of users that have this issue.
The next release of Firefox ESR is going to be version 17, due in Q4 of 2012.