The Portable Hole is described as:
This fine black cloth, soft as silk, is folded up to the dimensions of a handkerchief. It unfolds into a circular sheet 6 feet in diameter.
Once deployed, it is neither worn nor carried. Is it at risk from magical and/or nonmagical fire?
Maybe?
You have already pointed out the similar question about magical nets, where it quotes from the DMG:
Most magic items, other than potions and scrolls, have resistance to all damage.
This would indicate that the Portable Hole has at least resistance to fire, even if made of cloth.
I would say it is a DM decision as to whether an item with resistance to fire also means that something is not flammable (personally I would say not, as a portable hole is quite an extraordinary item so I would say it would need a more extraordinary means of disposal).
Also worth noting that, generally, a creature's equipment is not subject to damage unless specifically noted (to avoid player frustration if nothing else), so the hole would at least need to be unattended for the potential firestarter to try lighting it up.
The text about "It unfolds into a circular sheet" describes the properties in "cloth form", just so that you know how the cloth looks and works. So one could leave it around unattended but still cloth, and it would be subject to being set aflame (though it would have resistance to fire damage like all magical items do). But once one uses an action to place it unfolded on a surface and activate it, it becomes an extradimensional hole instead.
In general, effects would go through the hole, since, well, that's what a hole is. The only thing that's different about this hole is the magical ability to use an action to grasp the edges and turn it back into cloth again.
