How to edit 'etc/hosts' file in non-rooted phone?

Unless I missed something, that is not possible. Without root access only the mechanism of flashing an update (via recovery or OTA) can modify /system. To be accepted, such an update archive must be signed with the same key as the ROM – which is something only the developer of the ROM (here: the LOS team) can do.

There might be a way via a custom recovery which lets you edit all file systems directly (see e.g. twrp). But without knowing what device it is, it cannot be said if a matching recovery exists (you can check that in the Devices section of their website for official images, and search XDA for unofficial ones). Apart from that, it's either rooting the device or not editing any file within /system.