The GPU on my mid-2011 27" iMac i7 died recently. Replacement for the 2 gig AMD Radeon HD 6970M 2048 MB card was around $840 locally, which represents around 35-40% of a new mid-2017 27" iMac i7. Although I had no issues with data lose (due to iCloud and Dropbox), I wanted access to my hard drive to at least erase it. I had read in other blogs that it is possible to boot with the integrated Intel graphics chip by eliminating the AMD/ATI extensions within the Systems library. Due to the addition of System Integrated Protection (SPI), using the Unix commands at the Single User Mode was not possible to move or delete the extensions in macOS Sierra, so the last resort was to directly access the HD using either a targeted drive boot with another Mac or use an external boot drive. I do not own a Thunderbolt cable, so i opted for the external boot drive, which I created with my MacBook Air. Here are the steps that I used to successfully reboot my iMac with the internal HD on the integrated Intel chip:
1. Create a boot disk using a different Mac: https://support.apple.com/en-us/HT202796
2. After creation, reboot your Mac (not the bad GPU Mac) and access the new boot drive. Under the folder System/Library/Extension, move or delete any file/folder that begins with either AMD or ATI. I elected to transfer them to a folder I created called DisabledExtensions.
3. Boot the discrepant iMac with the new boot drive by pressing the option key during the startup. This gives access to choose the startup drive.
4. Once fully into your iMac, log on to your HD (if required) and in the System/Library/Extension folder, either move or delete all references that begin with AMD or ATI.
5. Reboot the iMac. Hopefully, this will allow you to operate on the integrated chip. There will be some graphic issues, but, if all goes well, overall you will have full access to your data, as I did.
I abbreviated the steps with the idea that anyone following this blog will know I have shortcut some of the detail in creating boot disks and logging onto a system from an outside peripheral. However, it worked for me.