My iPhone 6 worked like a champ for almost 30 months, but eventually the daily charge/discharge cycle took its toll, and I began to have the experience of the battery not lasting me through a full day.
Having damaged an earlier phone when opening it up, I shelled out for the iSclack case opener for my first foray inside the iPhone 6, and opening up was dead simple.
I thought I could get away without detaching the screen, but my initial attempt to pull out the connector-side adhesive strip ended with the strip breaking about a quarter of the way through. Then I was faced into an altogether uglier and more forceful process of prying up the battery in order to grab the remaining part of the strip, and completely disconnecting the screen seemed prudent. I was a bit nervous when the old battery began to bend under the force, but eventually I got a good hold on the strip and pulled it out, and the old battery was released.
I carefully followed the very clear instructions on attaching the new adhesive strips to the new battery, and installation/reassembly went very smoothly.
It seems like part of the plastic frame insert got caught beneath the screen on reassembly (see photo); I might be able to release it if I opened it up again, but it doesn't seem worth the effort.
It's tricky to get the technique for removing the adhesive strips. Keep moving your finger down the strip you're pulling out, so you're never more than an inch from the device, and make sure the part you're pulling out stays wide and flat and doesn't contract into a single thread.