WIll a hero with damage over time die if he is banished?

Solution 1:

No, being banished (Astral Emprisonnement, Disruption, Phase Shift etc.) makes your hero invulnerable.

The only exception to this rule is Sanity's Eclipse (which does not deal damage over time) that can damage you while you are Astral Emprisonned.

Being Banished in any form does not pause a debuff timer (like a DOT or a stun) which allow you to negate some part of the damage when you are banished.

Solution 2:

Banish makes a target invulnerable, except from the hero that banished them. Sanity's Eclipse works through Astral Imprisonment, and Soul Catcher, Shadow Poison and Demonic Purge all affect targets within Disruption.

Phase Shift makes you immune to everything.

A Damage over time spell will continue to tick while banished, but do no damage. e.g. A hero that is Doomed and has 3 more seconds of it left will not take any damage if he is Astraled. If 6 seconds left, he will take only 2 ticks of damage, assuming Astral Imprisonment is at max level.