Can I take the screen out, without turning it off?

Yes you can but there is a risk of you hurting yourself (small electric shock) or a greater risk of the phone being damaged with contact of water since the phone is on. Just make sure that the exposed motherboard does not get into contact with anything so that no extra damage is done

From my experience I want to say that it turned OFF because the battery discharged, this happened to me a couple of times, but after charging phone with either the lighting cable or a battery-case I noticed it still worked.

Now, the advice to you (if you ever do it again and you have no way to turn OFF the iphone) or to someone else who cannot turn OFF the iPhone 5 before opening it or changing the screen is that:

You do it REALLY careful, and you do the process of opening it the proper way (no more than 90degrees) then take out the little lid covered by the three(3) screws which covers the cables from the screen.

After you are done with this, then you should move to the battery connector on the lower part, take out the two(2) bottom screws, then disconnect the battery from the logic board.

Then you know that the phone is OFF, and don't forget to connect the battery and place all the lids with their proper screws after you are done working on it and before you close it.

For more help click on this Youtube video that I found informative with the answer.

https://www.youtube.com/watch?v=FBIxr8W7...

I have simular problem. I dissamble phone cause wanted to replace camera, but forgot tu turn off the phone first -1gen SE. I separted screen form phone . Assambled everything back and now phone is not working. Doesnt want to turn on. Is it possible to damage your phone if you remove connections from inside the phone without turning it off first. I think there were 4 connections in my case.

i was changing a screen and had this problem on an iphone 11, and was confused on what to do cause it was power looping and the screen that wasnt respontding to touch cause the screen was faulty so i went to a cool unhumid room and disconected the ribons and changed the front camera and speaker to a new screen then connected it and hoped for the best and it worked but was risky.