Ryan,
You should be able to use the System Cache and GZip plugins once we figure out the issue.
We have made some progress, and now I want to throw a few questions back to you. The problem isn't IE. The problem is simply a caching issue with all browsers, but IE handles it the worst, by far. Other browsers, depending on how you've navigated your site and the Facebook Page Tab, will simply be inconsistent.
Do you know how your website and Apache are configured? I don't want to get too technical, but will explain the basics of what we're seeing. Of course, this may be 101 for you

* A user visits a page on your site
* Your server responds with the HTML for that page along with a 'header' that says the page will expire in 24 hours.
* If that page is visited again, the browser will see that header and simple re-use what it has already fetched
For JFBConnect, that's 'bad' as we need to modify the page's HTML a little based on where the page is being viewed. However, since it's being cached by the browser, the 'first' place it's viewed is what's always loaded.. and so we can't inject some of our special sauce to undo that.
Now, this is all something we'll be trying to fix.. Until then though, if you can check, mainly your .htaccess file, to see if you have a statement like below:
ExpiresDefault "access plus 1 day"
Or anything else that says "expire" frankly, can you let us know? It may be best to remove that statement and try to test again. You'll still be able to do all the caching you want server-side (GZip, minify, system cache, etc), but on the client side, that seems to be the problem.
If none of this makes, let me know. Again, we'll still be looking into a more fool-proof solution that will work with client side caching, but that may take some time.
Thanks again for your patience, and we'll continue to do whatever we can to get this to work!
Alex