Problem

‘The game has crashed. Please try to reconnect.’ immediately after champion select.

Solution

Open /etc/hosts on the host computer in the text editor of your choice.
On the line that resembles:
127.0.0.1   localhost.localdomain   localhost
add the name of your computer (found in /etc/hostname). For example:
127.0.0.1   localhost.localdomain   localhost   NEXUS

Problem

‘Unable to connect to server. If you have a firewall, it may be blocking the connection. Please refer to your firewall's documentation. Would you like to retry the connection? [Retry] [Cancel]’ after champion select.

Solution

Open winecfg (if on PlayOnLinux, use Configure -> Wine -> Configure Wine). Go to Libraries. Add the override dnsapi (builtin, native).