How are Fiends, Celestials, dragons, etc affected by the antimagic field spell?

I assume they can walk into them. Do they lose abilities, take damage, etc?

Just the same as every other creature: they can't use magic or magical abilities

Antimagic field prevents the use of magic and magical abilities in a variety of ways within its area.

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.

Nothing is special about any of these creatures that give it any special interaction with an anti-magic field. Dragons can still fly and breath fire in an AMF for example, but a spellcasting dragon would not be able to cast those spells inside the field. Any magical abilities the creatures have will be suppressed as well as any magical effects present on them.

None of the creature types are inherently magical in a way such that they would be affected separately from any magical abilities they have. There's no such thing as an inherently magical creature type, for example.

As always, we can use the test provided in the Sage Advice Compendium to tell what a magical feature is:

Determining whether a game feature is magical is straightforward. Ask yourself these questions about the feature:

• Is it a magic item?


• Is it a spell? Or does it let you create the effects of a spell
  that’s mentioned in its description?


• Is it a spell attack?


• Is it fueled by the use of spell slots?


• Does its description say it’s magical?

If your answer to any of those questions is yes, the feature is magical.

If it doesn't fall into at least one of those categories, it is not affected.

So, the only an antimagic field has on a creature (the type does not matter) is the suppression of magical features, abilities, and effects on the creature. They take no damage or other effects.

Generally speaking, they just lose access to any magical effects on them and can't cast spells or use any other magical abilities, the same as every other creature in the game. From the antimagic field spell description:

Creatures and Objects. A creature or object summoned or created by magic temporarily winks out of existence in the sphere. Such a creature instantly reappears once the space the creature occupied is no longer within the sphere.

Fiends and celestials usually aren't native to any of the D&D worlds and are only native to their home planes. If a fiend or celestial (or fey, elemental etc.) are only present because of a summoning spell then they will vanish until the space they occupied is no longer inside of an antimagic field.

Do they lose abilities?

Nominally speaking, no, but it depends on what kinds of abilities we're talking about.

For example: A Dragon would not lose its ability to breathe fire. We recognize that Dragons are inherently magical, but their ability to breathe fire is the inherent magic of the universe itself, and not suppressed by an Antimagic Field.

Conversely, many Dragons/Fiends/Celestials have innate Spellcasting abilities. These abilities would be suppressed in an Antimagic Field, because that's specifically what Antimagic Field says it does (at least in the spell description):

[...]

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.

—Antimagic Field, PHB, pg. 213

Do make sure to check that the entity in question isn't literally a diety/god (because they and their features are explicitly unaffected by an Antimagic Field) but otherwise, spellcasting would be suppressed.

Other features may or may not qualify depending on their nature. Things like the Devil's Sight feature common to Devils would not be suppressed, but a feature like the Magical Weapons feature that Erinyes (Monster Manual, pg. 73) have would be suppressed because the weapons are expressly described as being magical, and Antimagic Field is expressly described as turning Magical Weapons→Mundane.

Do they take damage?

Unless the creature's statblock specifically says that they take damage or are otherwise harmed in the presence of an Antimagic Field, they don't take damage. Some such creatures might be summoned by a spell, and in that case, they would temporarily disappear for the duration of the field:

Creatures and Objects. A creature or object summoned or created by magic temporarily winks out of existence in the sphere. Such a creature instantly reappears once the space the creature occupied is no longer within the sphere.

—Antimagic Field, PHB, pg. 213