What happens if you are holding an Iron Flask with a demon inside and walk into an Antimagic Field?

Do the magical powers of the Iron Flask stop, releasing the demon? Also, would the demon then be sent back to its original plane (the Abyss)?

It stays in the flask

A related question:
What happens to a bag of holding in an Antimagic Field?

The iron flask seems to function like a pocket dimension for the captured creature, which is why I'd personally rule that the flask simply turns into a normal flask, and is cut off from the pocket dimension for the duration.

The Antimagic Field spell itself seems to confirm this (emphasis mine):

Spells and other magical effects, except those created by an artifact or a deity, are suppressed in the sphere and can't protrude into it. A slot expended to cast a suppressed spell is consumed. While an effect is suppressed, it doesn't function, but the time it spends suppressed counts against its duration.

The word suppressed is not the same as dispelled. This means its effects simply cannot be used, not that the effects disappear. This is especially true if you look at the last sentence, which specifically states that any spell cast is "active", it just doesn't do anything.

I'm breaking it down into parts, for clarity.

Do the magical powers of the Iron Flask stop

Yes, that's what Antimagic field does; it stops items from functioning.

releasing the demon?

This is unclear and depends on how exactly your Iron Flask works. Since the demon probably doesn't actually fit inside the flask, it must either be magically reduced in size to fit or be banished to an extradimensional space linked to the flask.

Given that you need to cast Identify to find out if anything is inside the flask, it sounds most logical to me that the creature isn't actually in the flask, but is in a separate dimension. Otherwise, you'd be able to determine if anything was in the flask pretty easily by shaking it.

In that case, it's likely the demon would not be released. And, since returning the demon is part of the magic of unstoppering the flask, likely it would not even come out of the flask if you pulled the stopper while in the anti-magic field.

If you instead decide that the creature is physically in the flask and being shrunk by it, then stepping into an anti-magic field would break the containment & shrink effects, and it would grow back too full size while inside the flask. This would probably destroy the flask or kill the creature.

Also, would the demon then be sent back to its original plane (the Abyss)?

No, nothing about the flask says anything about returning contents to their original plane, and the demon coming from the flask isn't a summoned creature. So the demon would simply remain wherever it appeared. This is also what happens normally, when you release the creature. Unless it has some means of moving itself between planes, it remains trapped where it is.

final note

if you rule that the creature is released when you step into the anti-magic field, or otherwise release the creature before stepping into the anti-magic field, do keep in mind that the Anti-magic field will also suppress the part of the Iron Flask that makes the creature friendly to you. So you'll probably have a pretty angry demon to deal with in that situation.

I'm going to go with the opposite answer and say that

It is forced out of the flask

Other answers assume that the flask works merely as a gate to an extradimensional storage space. While not an unreasonable one, it is still a GM ruling, as nowhere in the description of Iron Flask does it say so. Instead, it says :

the target must succeed on a DC 17 Wisdom saving throw or be trapped in the flask.

and

Once trapped, a creature remains in the flask until released.

and also

You can use an action to remove the flask's stopper and release the creature the flask contains.

Taken literally, this shows that the creature actually goes inside the flask. Therefore the magic that holds it inside is suppressed when in the Antimagic field.