-midnight,
I access Facebook, without Java (in both IE and FF), and don't have any problems. While I believe some features of Facebook [such as some of their photo uploading routines (? ? ?)] may use java, the site, in general, does not.
I also access secure sites, such as my bank, credit cards, and brokerages, all withOUT java.
So I believe it much more likely that NoScript [with or without Java activated] is the culprit that's blocking both your Facebook and your online banking.
NoScript can block Java, JavaSCRIPT, Flash, and .PDF files --- to name just some of what it can block. Basically, you have to watch for NoScripts prompts, and "train it" as to what you'd want it to allow. Quoting from it's website:
When you install NoScript, [it defaults to blocking] JavaScript, Java, Flash, Silverlight and possibly other executable contents... You will be able to allow JavaScript/ or Java/... execution (scripts from now on) selectively, on the sites you trust. You can allow a site to run scripts temporarily, if you're just surfing randomly, or permanently, when you visit it often and you really trust it [like your bank]. This means that NoScript learns from your own browser habits and tends to disappear in the background after a while, but it promptly comes back to save your day if you stumble upon a malicious web page.