Is it possible to update a jailbroken iOS 7 to a version Apple has stopped signing? My iPad is running 7.0.2 and is jailbroken. It looks like Apple is not signing the 7.1.2 any longer.

Is there a way to upgrade iOS to 7.1.2 without letting iTunes contact its home (i.e. Apple's servers)?
Nope. It's impossible to upgrade iOS firmware without connecting to the Apple servers.

Currently, there is a jailbreak for iOS 8.1 using Pangu which is being signed at the moment. iOS 8.1.1 update which is due in a few weeks will patch the jailbreak. It could be the right moment to upgrade your iOS version and re-jailbreak it.  @saurik confirmed that the current iOS 8 jailbreak is stable.