Agree with SergZak - in my case there was definitely a hardware issue - I can see a huge difference in the performance of the new, properly functioning, phone. I think there was likely one little point - solder or otherwise - that was affected by heat and the initial burden of downloading the backup, plus all of the apps, plus all of the photos and video - brought the internal temperature up enough to create the issue.
Chrisemersonnc - this is, in my case, definitely a hardware issue - no amount of reboots will help you - however, in my case, I found that by pinching LIGHTLY on the upper right-hand corner, it stopped the volume constantly turning up and at least allowed me to use the phone. If you find the same symptoms - definitely contact Apple - they were quite helpful (once they had me go through all of the routine things - reboot, reset, restore, setup as new phone...) So prepare to spend a little time on the phone while they escalate you up the ladder.
Good luck!