A month or so ago, my phone started shutting itself off when displaying a charge of 70% or even 80%. I couldn't use it very long before it would turn off and I'd have to plug it in. I didn't want to pay ~$80 at the apple store to get the battery replaced, so I thought I'd do it myself.
When I tried pulling the first adhesive strip around the corner, it ended up breaking. I kept trying to get a grip on it again, but it eventually broke off even more and I couldn't get to it. For the second adhesive strip, I just pulled it straight out like in the video for the iPhone Battery Adhesive Strips Replacement guide, which worked great. I ended up having to heat up the back of my phone and pry the battery out. Luckily, the battery didn't bend more than slightly and I was able to get it out. Putting the new battery in was easy enough. My phone turned on fine, and I calibrated the battery like it said. It works great now and doesn't shut off!
I recommend pulling the adhesive strips straight out, rather than trying to pull them around the whole battery.