Does Apple's Out-of-Warranty Replacement cover iPhones regardless of condition? I have a water-damaged iPhone 6. I took it to a non-Apple repair shop after the plunge (the Apple Store was too far away for urgent service), but it still does not work. It powers on, but the screen is always black or only backlit. Overall, the phone is in very good condition cosmetically.

On to the question. The repair shop removed some heat shields from the logic board during the cleaning process. Can I still bring my iPhone to the Apple Store for an out-of-warranty replacement? The service is currently listed on their website for $299, but are there special conditions I should be aware of before making the trip? Will a technician open the phone for inspection before offering a replacement?
No, it does not. Out-of-warranty service is provided for phones which have not suffered what is defined as "catastrophic damage" and additional fees on top of the OOW service fee may be charged up to the replacement value of the product. 

Please review Section 3 of the ERS/OOW Terms page: Source

Note that in this response, I am not qualifying your repair as either OOW or beyond that and needing to pay replacement fees. That is just a potential outcome, and is meant to highlight that not all OOW damage is covered.