iOS7 & iPhone 5 Reboot Loop

I just had this issue. I left for a trip to Europe. At some point I felt the phone vibrate and took it out to find it in the same reset loop that you are describing. When I got back to the hotel I plugged it into a wall and it actually breifly booted up, but with no networking (3g or wifi/bluetooth). After staying on for about a minute or so it went back to the reset loop. Obviously being overseas, I had no access to a computer to sync with, so I was stuck without phone coverage for the rest of the trip (thanks Apple!).
Just got home and tried to resync with iTunes to do a restore. Failed just like it did for you. However.....
I think I may have just fixed the problem (at least temporarily). I opened the iPhone and disconnected the battery temporarily. When I reconnected it my iphone could boot up again successfully. It now seems to be working fine. It's only been about an hour though, so time will tell if I have to deal with this again.

I had exactly the same problem - in the middle of web browsing. The phone turned off on its own, then got into this reboot loop. Reboot never passes apple logo.

First tried reboot with on/off + home botton. Unresponsive

Then tried recovery mode retore, using iTune. This is a bit tricky. It took me several tries to get it done correctly - you need to hold the home button first, then, while keep holidng, connect USB, until you see the iTune logo.

Then, iTune will ask you whether you want to restore. I clicked yes, and the restore started. Normally, this should have fixed the problem (according to Apple Genius Bar tech).

But in my case, the "restore" didn't work. In the end, I had to take my iphone to Apple Store. People there could not fix my problem either. They ended up giving me a brand new phone (since my old phone is only 3 weeks old and still under warranty). They said they have seen the problem before, typically can be fixed by restore. In my case, though, it might be a hardware issue???

I'm having the exact same issue. It was doing it on my iPhone 5 and my iPad as well. I went to apple store, they did iTunes reset and it stopped, but then it started up again. I ended up upgrading my phone to a brand new iPhone 5s and now it is doing it on it as well! We have lotus notes and mobileiron so I think that issue may have to do with that if others are having issues with exchange. Anyone have mobileiron on their phone that is having this issue?

I was able to get it out of the DFU loop.

1. With the phone plugged into a wall outlet hold the sleep/off bottom and the home button for 10 secs. this will stop it from looping. Release

2. Next hold the home button for an additional 10 secs. This put it into recovery mode (Cable with the apple logo).

3. From here I was able to complete my restore in itunes.

Thanks! Saved me from sending my phone into repair!!!