Endless restarting and flashing on Apple logo loop

If you are using DFU mode correctly (it's tricky; you must follow the instructions EXACTLY) and it doesn't go into Recovery mode the phone's hardware has failed or the phone has been incorrectly jailbroken. There is nothing further you can do.

DFU mode (with prep steps):

• Install the latest version of iTunes on your computer, making sure you use a computer that has administrator privileges.
• Reboot your computer.
• Disable your antivirus and firewall.
• Launch iTunes.
• Connect the USB cord to a USB port directly on the computer (not a USB hub). Do not connect to the phone.
• On the phone hold the HOME and SLEEP buttons at the same time for exactly 10 seconds, then release the SLEEP button but continue holding the HOME button. If you see an Apple logo appear you held both buttons for too long. Start over.
• While still holding the HOME button connect the dock cable to the phone.
• The phone should appear in iTunes in Recovery mode. This may take up to a minute. When it appears release the HOME button and the phone should restore.

As I posted to another similar question, I got this problem fixed followingly (is that a word?):

For those like myself who don't backup your device regularly and fear you may lose data, there is hope. My 4S started looping the apple logo at start up and wouldn't go anywhere from there, the screen went black and then the apple logo would appear again over and over... Browsing around I found that recovery mode and resetting the phone was suggested, some suggested downloading software to fix the issue but I found that rather dodgy so I decided to try something else.

Now I must remind you that opening your phone will void your warranty or such if still applicable, my phone's maybe four three years old and I've replaced the screen myself so I figured I'd try something to shock the phone back into life somehow (think the last popup reminder said I hadn't backed up in 29 weeks so meh).

What I did was opened up the back cover and removed the battery, plugged it back into the charger (other end in my mac) but the loop would still occur. I plugged and unplugged a few times, unplugging right after the apple appeared, hoping for this to "confuse" the phone which was so stubbornly stuck in "recovery mode" (from what I've understood through other threads). Notice that the phone did appear to turn on without the battery connected, but plugged into a power source (not sure if that should happen since after dead battery you need to charge a while until the phone wakes up), just something mildly interesting I picked up on.

This did not appear to have any effect tho, but I decided to pop the battery back in its place (I did leave out the little black plate that goes between the battery connector and the motherboard, covering the antenna cables connecting point, not sure if that had any effect or how it would effect fully working phones (I only use this phone as an "ipod" since the sim reader has been dead for some time, but I still use the camera and music)). Pressed the power button and noticed the apple logo stayed for longer than before (became hopeful!) and eventually made way for my home-screen!

That's how I got it working, don't know if you can do a similar thing with other models. And sorry for rambling on like that, that's just how I express myself and I can't afford to hire an editor to work my internet posts. But I can leave a TL;DR version.

TL;DR Iphone 4s stuck on logo loop? disconnect battery and do magic.

Here is the Apple tech note on error 4005: iOS: Restore errors 4005, 4013, and 4014 - Apple Support

Hi, I think you need to take a look at this article: iOS: Restore errors 4005, 4013, and 4014 - Apple Support

There are two options I can suggest:

1. Wait until the battery dies and then attempt to re charge and reboot OR

2. Bring to an Apple store

Lawrence gave the OP that article about 4 posts ago -

Is there a reason that almost every post you make duplicates a post already made in a thread?