Phones will suspend and kill applications that aren't actively used; with iOS this has always been the case (you can't have ANYTHING running in the background on iOS unless it's navigation, calls, or music, that is you need to request special exemption from it). It might be possible on Android but even there the phone will slumber while in your pocket.

Phones are not servers.

Phones turn off their radio when the screen is off and go into a sleep state (waking up in intervals). Keeping the radio on 24/7 is just going to increase power usage by a lot. It's not the CPU that's the concern.

Even ignoring this issue, it would open your phone up to getting DDoSed by strangers on the internet.

How long delay does this sleep state impact messenger applications? Are we talking seconds or milliseconds?