iPhone turns off when removing case

It sounds like something may be shorting to ground, or some other electrical line when under pressure (assuming there is some flexing and struggling when removing the case). Most likely something is getting squished into something else it isn't supposed to be. Could be any number of things, but I will specifically call out that the main power rail for this phone does go to the display, so this could be as simple as the cable for the display is getting pinched, or was damaged.

If the battery was changed recently I would take it back to the person who did the repair and let them know since it sounds like this was something that started as a result of the repair.