Error 2001 and 2006 keep appearing when I try to restore

Sam Coldman, these could be the result of corrosion damage or shorted components. I suggest that you no longer try to turn it on or try to charge it. Disassemble it using this guide and properly clean it. Use this guide to clean it. It was written for a 3G but all the points are still valid for your phone as well. Always check your board for any missing, burned or otherwise damaged looking components. you said you tried to replace the battery, but did not tell us if you did. Definitely make sure that you have replaced the battery, reassemble your phone and re-evaluate for any problems. The heating up could be caused by a shorted charging coil. Check the voltage on the battery connector (with the battery disconnected)and see what you get when the phone is plugged into either a USB port or an AC charger. If it does not charge properly, you may consider another battery,a new dock connector (of course you are using a compliant cable :). If none of that works for you, the issue will be either with the charging coil or the power management IC. Here are a few items you want to check as well. I hate to refer to those as fuses, but they most certainly control part of the charging circuit. First one is reference designator L3, it is a FERR-120-OHM-1.5A inductor. L4, right below it , has the identical value. Next would be the charger inductor L28 2.2UH-1.8A-155MOHM. All of these are pretty much available at mouser. com or digikey.com.

Of course, you also want to try what Apple suggest:" If you experience this issue on a Mac, disconnect third-party devices, hubs, spare cables, displays, reset the SMC, and then try to restore. If you are using a Windows computer, remove all USB devices and spare cables other than your keyboard, mouse, and the device, restart the computer, and try to restore. If that does not resolve the issue, try the USB issue-resolution steps and articles listed for Error 1604 above. If the issue persists, it may be related to conflicting security software. If the errors persist on another computer and known-good USB cable, the device may need service." from here.

Last but not least, or may be it should be first;) try a hard reset as described 1) Turn off the iPhone. Make sure the USB male end of the cable is attached to the system. iTunes should be launched.

2) While holding the home button plug your iPhone in.

3) Follow the on screen instructions to a hard reset. This removes all data from the phone.

4) Try to restore your phone. Hope this helps, good luck.