How to backup a locked iPhone?
If you've lost the password/passcode to a locked iPhone, you have very few options:


If you have access to iCloud, remote wipe it from there
Put the device in recovery or DFU mode and reinstall your preferred version of the iOS.


Obviously you'll lose all the data on the device -- without access to the device you can't authorise iTunes to access the contents to bake a backup, for example - and you're limited to the version of iOS that Apple is still currently signing (they typically stop signing older versions a few weeks after a new version is deployed)

Sadly, you're quite stuck, and apart from the above, or a visit to your local Apple distributor, there isn't much you can do.

If, however, your phone is stuck in a boot loop, you can try one of the few "hail Mary" options which have saved my bacon more than once. There's a tool called Recboot which will try to put your phone into recovery mode (or you can try to octopus-like method yourself), or (and this is the life-saving bit) get you out of recover mode.

I don't personally own the software, but have used it to get out of a few jams in the past. You can find instructions on using the (free) software, plus download links here