Chosen Solution
The Genius Bar says that I have to replace my phone, because after replacing an unresponsive display, it still won’t work. I have a bunch of important stuff on there from the past week since it was backed up (graduation pictures, etc). It connects to iTunes, but I can’t unlock the phone. Is there a way to unlock my phone remotely or to recover what is on it?
Other than changing screens or batteries, the Apple “geniuses” can’t really do much to help you. You need to find a shop that does micro-soldering repairs as they may be able to troubleshoot the problem. The only way to get the data off the phone is to get it work enough to fully connect to iTunes (unlocking the phone and trusting the computer). The phone may be fully repairable or it may only be recoverable.