How do I fix iTunes' error 3194 when upgrading iOS? My iPhone is on iOS 4.2.1 and I want to upgrade it to iOS 4.3.3. I know there are newer iOS versions available but they can't be jailbroken untethered. 

I have the 4.3.3 package from Apple, and I know I can shift-click the "Upgrade" button in iTunes to select this file. When I do, iTunes first extracts the file (no problem) and then tries to verify it, and here's where the trouble starts. iTunes gives me an error 3194 which according to a Google search means that iTunes didn't get a useful response from the validation server. 

I've checked for these possible causes:


The hosts file might have a custom entry which prevents iTunes from getting a response.  I've checked: There are no entries in the hosts file, only #comment lines.
A tool like tinyumbrella might have changed the lookup. I installed tinyumbrella and checked: it's set to use normal iTunes settings, no overrides are active.


How can I get rid of this error and upgrade to iOS version 4.3.3? I'm on Windows.
That specific error is clearly spelled out on the general iOS error charts - http://support.apple.com/kb/TS3694
Saying that it didn't get a useful response is a bit vague.
iTunes requires a specific, signed, affirmative response from the apple servers before it will proceed. This is clearly by design. Additionally it checks the version of your iTunes and is asking you to get a newer version.
You may not want to upgrade iTunes as some people dislike some of the changes, but it looks like that's the cause here assuming you really removed all the umbrella changes properly. It also might ask for a newer iTunes but the problem isn't actually that.
There is a whole set of things to check at the top of the main article: security software, iTunes store tests, using old or modified .ipsw files.
It might be good to set up a new user on the windows os and see if you can download 4.3.3 cleanly (or find a pristine copy on the intertubes) and try the restore and debugging of the hosts file from a cleaner start.