I leave all my Macs on all day. Most are laptops with the lid closed, but two are desktops. All plugged in. They just sleep.
I never turn my phone off. It's plugged in at night though. On occasion I restart it.
My iPhone is my only phone and my wife's iPhone is her only phone. We do not have a landline at home. So, just like a landline our phones are always on. You don't want to miss an urgent or emergency call at 3am because you turned your phone off.
Anyway, I have never had any issues whatsoever. Did this with our phones before the iPhone and the phones before that. My own iPhone has been on since November of 2012 when Apple replaced the previous one. I still have 89% of my original battery capacity which I think is pretty good.
That app looks very similar to one i already use called Presence. That is also a sort of security\monitoring camera app. I used it on my old 3GS plugged in and set up in the kitchen once when we went on holiday so the app was running with the phone on and active for about 9 days without an issue and then i've also used it on my iPad for keeping an eye on our dog when it was younger to make sure he was ok whilst i was at work.
Other than having to ensure the device was plugged in with a charger to stop the battery running out i've had no issues with either device. Might get a bit warm but thats it. Not sure if this app has it but Presence has an option to disable the screen on the device thats acting as the camera as you dont really need it on plus it stops it giving out a glare when its dark.
I use my iPad sometimes for Presence Pro which is similar to Manything. I have a stereo that has a charging dock for the iPad that handily is the right angle and position to use it to keep an eye on the dog if i need to. I've used the app for years now on both the iPad and my old 3GS as a camera and never been an issue.
Only thing you need to sort is make sure they have power to stay on all the time.