When do support and other 0 hit point cards get destroyed? Do they die when an undefended zone is attacked and all units are destroyed? Are they destroyed when the zone is burning?

They are never destroyed unless your opponent plays a card that destroy theim.

What do you mean about 0 hit point?

Having 0 hit point is different from having no hit point.

I mean if you are talking about a unit, if a unit has 0 hit point the unit is destroyed.

As Shindulus explained. See p. 7 of the rules:

Hit Points: The amount of damage a unit can
take. If a unit has as many damage tokens as it has
hit points, that unit is destroyed and is placed in its
owner’s discard pile.

This is of course valid for any number of hit points, even 0. If hit points are 0 (e.g. through an action played by HE which reduces the HP of a unit by -1 from 1 to 0) and there is no damage (so damage = 0), as per the rules, the unit (or legend) is destroyed. Support cards do not have hit points, so this does not apply to them.

If you mean: how can I remove cards that do not have any hitpoints, then you can do this only via card effects like destroy a support card. The burning of a zone does not affect the cards located in that zone, they still contribute their power, can play actions, defend, attack (if in battlezone) etc.

Thanks for both your answers, that is clear now.

I should make 2 small corrections to my post:

1. Cards that reduce hit points are common for DE, not HE

2. If the hit point reduction targets a unit, a legend would not be affected, since it is not a unit