Does the War Caster feat, which gives advantage on concentration checks, also apply to the Trickery Cleric's Invoke Duplicity feature?

Yes, Invoke Duplicity uses the same concentration mechanics as spells

Because Invoke Duplicity says "as if you were concentrating on a spell", this means that any features affecting the mechanics of concentrating on a spell also apply while concentrating on Invoke Duplicity. This includes War Caster, which applies to "Constitution saving throws that you make to maintain your concentration on a spell when you take damage".

The language is ambiguous, the DM will need to adjudicate

Invoke Duplicity says:

As an action, you create a perfect illusion of yourself that lasts for 1 minute, or until you lose your concentration (as if you were concentrating on a spell).

War Caster says:

You have advantage on Constitution saving throws that you make to maintain your concentration on a spell when you take damage.

This can be read as:

1. "Treat Invoke Duplicity like it's a spell", and so War Caster applies

or

2. "Make concentration saving throws like Invoke Duplicity is a spell", but it isn't, so War Caster does not apply.

Different people, will, with the best of intentions, read this differently

See answers by Ryan C. Thompson and Eddymage that argue these two points of view.

So the DM adjudicates

You can read the arguments and parse the language, and maybe one interpretation seems clearer to you. If so, great, go for it!

If not, just make a decision, and move on.

I'd like to make a case that the rules are not sacrosanct. They are not there to build an edifice on the internet so that every player is following a blessed interpretation.

The rules are there to give each game a place to start. The rules themselves say this when in multiple places they talk about the DM being master of rules.

It is less important that the DM come up with an answer that is perfect compared to a bunch of other people on the internet than it is to come up with an answer that works for your game.

My advice

Likely, the ruling that makes the most sense is to treat Invoke Duplicity like a spell and allow War Caster to give the saving throw a benefit.

I mean, really, how cool is that? It's not an automatic win, it doesn't guarantee the cleric won't drop concentration, but it gives them a real advantage.

In my experience, this is likely to result in some pretty exciting battlefield moments. I played a cleric in an Eberron campaign to 20th. My cleric worked like hell over the levels to raise constitution, until the last few levels, they had a 20. It was really fun. There were battles where I had upcast Banishment and banished several opponents, and took concentrated fire and mostly managed to keep concentration, making save after save. It was equally dramatic finally losing concentration and having two fresh opponents show up.

Bottom line, this my advice. If parsing a rule like this, even after deep scrutiny, it is unclear to you how different aspects of the rules interact, rely less on squinting to interpret one way or another, and more on what will make the game fun.

No. Concentrating on something as if it were a spell does not make it a spell. That doesn't mean you can't do it, but it has implications.

You have advantage on saving throws to maintain concentration on spells (when you take damage). You do not have advantage on saving throws to maintain concentration without qualifications, and concentrating as if you were concentrating on a spell does not qualify the thing itself as a spell, it qualifies that you can fail a concentration check and lose the thing you're attempting to keep.

You make concentration checks for Duplicity as if it were a spell; this is a requirement, not a benefit. Contrast against other abilities (such as bardic inspiration) which do not require concentration to maintain their benefit; the only difference here is that if you get hit, you stand to lose your duplicate.

Nothing in the line suggests that because you must pass a concentration save (like you would with a spell) when taking damage that the ability actually becomes a spell, so nothing in the line suggests that you would apply the benefit of War Caster.

There are no abilities in the game that say "as if you were concentrating." They all say "... on a spell," because the rules for concentration exist in the Spellcasting section of chapter 10. Concentration is always either on a spell, or as if concentrating on a spell, because there is no other reference point for concentration rules in the game.

You can do anything you want in the game that's fun for everyone involved, but to make this consistent you'd have to do it for every ability that says "as if you were concentrating on a spell." I can't claim to know any of them off the top of my head except for the one you've referenced here, but anyone else with such an ability would be reasonable in expecting it to apply to theirs, as well. Whether or not that impacts the game in a good or bad way is up to the DM.

A strict reading does not allow using this benefit for Invoke Duplicity.

The text of the War Caster feat says:

You have practiced casting spells in the midst of combat, learning techniques that grant you the following benefits: [...]

It particularly refers to spell casting, whilst Invoke Duplicity is a Channel Divinity feature:

Channel Divinity

At 2nd level, you gain the ability to channel divine energy directly from your deity, using that energy to fuel magical effects. [...]

It refers to magical effects, not spells. The Invoke Duplicity's bit

you create a perfect illusion of yourself that lasts for 1 minute, or until you lose your concentration (as if you were concentrating on a spell)

provides the game mechanics under which you can maintain the illusion: in particular, such mechanics follows the concentration rules for spell, but since it is not a spell War Caster does not provide any benefit.

Other game features use the same text:

• Sorcerer Draconic Presence:

[...] As an action, you can spend 5 sorcery points to draw on this power and exude an aura of awe or fear (your choice) to a distance of 60 feet. For 1 minute or until you lose your concentration (as if you were casting a concentration spell), each hostile creature that starts its turn in this aura must succeed on a Wisdom saving throw [...]

• Warlock's Dark Delirium:

[...] As an action, choose a creature that you can see within 60 feet of you. [...] On a failed save, it is charmed or frightened by you (your choice) for 1 minute or until your concentration is broken (as if you are concentrating on a spell).

• Wizard's Minor Alchemy:

You perform a special alchemical procedure on one object composed entirely of wood, stone (but not a gemstone), iron, copper, or silver, transforming it into a different one of those materials. For each 10 minutes you spend performing the procedure, you can transform up to 1 cubic foot of material. After 1 hour, or until you lose your concentration (as if you were concentrating on a spell), the material reverts to its original substance.

These are other examples where the intent of the text "as if you were concentrating on a spell" is to redirect the reader to the concentration rules: it is not saying that the feature is equivalent to a spell.

It is not game breaking if a DM allows it.

Since the mechanics is the very same, and it is still related to an ability that "fuels magical effects", a DM could reasonably rule that War Caster's benefit applies to the Invoke Duplicity's concentration saving throw. Since the number of uses of Channel Divinity is limited between rests (just one per rest until 6th level), this ruling is also balanced from a game-breaking point of view.

Both readings are compatible with RAW, but it makes more sense for War Caster to apply. There are weird inconsistencies if you rule it doesn't.

The question boils down to how strong you read the "as if": whether it makes concentration on Invoke Duplicity and other such abilities subject to other rules which apply to concentration on spells. Squinting harder as the "as if" isn't going to clear it up. We can look for narrative inconsistencies that follow from either choice.

Nobody is suggesting that Invoke Duplicity and similar abilities¹ fully count as spells. They can't be counterspelled because activating them isn't casting a spell. They can't be dispelled because the magical effects they create aren't created by a spell, and per Sage Advice, Dispel Magic is strictly limited to effects of spells, not stuff like Wild Shape. The argument here is only that concentration on them follows any and all rules which apply to concentration on spells, because that's what "as if concentrating on a spell" means.

If concentration on Invoke Duplicity et al is the same as concentrating on a spell, then War Caster applies when the concentration save is due to damage. And Sleet Storm and Earthquake can both break concentration without doing damage. (War Caster doesn't give advantage in that case; it's limited to concentration saves for damage.) And a raging barbarian can't concentrate on these abilities, the same as they can't maintain concentration on spells. (Rage doesn't prevent activating such abilities because they aren't spells you cast, but it does prevent maintaining concentration so only the up-front effects, if any, could happen.)

But if you decide the other way, you run into weird narrative inconsistencies, like that Sleet Storm can only break concentration on spells, not spell-like abilities which also require concentration "as if on a spell":

If a creature starts its turn in the spell's area and is concentrating on a spell, the creature must make a successful Constitution saving throw against your spell save DC or lose concentration."

Earthquake is pretty similar but doesn't say "on a spell", so does always apply. It seems very inconsistent if it can break concentration on spell-like abilities but Sleet Storm couldn't.

The ground in the area becomes difficult terrain. Each creature on the ground that is concentrating must make a Constitution saving throw. On a failed save, the creature's concentration is broken.

Barbarian rage:

If you are able to cast spells, you can't cast them or concentrate on them while raging.

You would be able to maintain concentration on Invoke Duplicity or similar abilities while raging if you decide that only the rules in the "concentration on spells" section of the PHB apply to them, not other effects that refer to concentration on spells.

@Eddymage's answer + comments assert that this is the only valid reading of the rules text, whether it makes sense or not that a sleet storm spell can't disrupt concentration on Invoke Duplicity, but a natural storm could (since this text appears in the Concentration section of the PHB "The DM might also decide that certain environmental phenomena, such as a wave crashing over you while you're on a storm-tossed ship, require you to succeed on a DC 10 Constitution saving throw to maintain concentration on a spell.")

I strongly disagree. It seems to me obvious that both interpretations are consistent with the "as if concentrating on a spell" language, and that having other things like Rage, Sleet Storm, and War Caster all apply makes a lot more sense.

(@Eddymage does agree that allowing War Caster and so on to apply is a good way to actually play the game, despite disagreeing that this is compatible with the rules text.)

I suppose one could even pick and choose on some other arbitrary basis which effects interact or not, e.g. that Rage blocks concentration on anything, not just spells, but that War Caster's benefit doesn't apply to concentration on non-spells. There's no justification for that in the rules themselves, so at that point it would just be the DM making rulings for their own table, which is fine but unless there's a problem you're fixing, it's better to be consistent with how you interpret different occurrences of the same rule interaction.

2024 removes ambiguity, with a formal definition of concentration which other things can reference. 2024 War Caster is: "Concentration. You have Advantage on Constitution saving throws that you make to maintain Concentration." (This also removes the qualification that it's only from taking damage, simplifying the benefit as well as making it stronger, at least for tables which noticed that wrinkle in the 2014 ruling.)

Footnote 1:
Eddymage found multiple similar abilities which say "as if you were concentrating on a spell", including:

• Transmutation wizard Minor Alchemy (1 hour)
• Trickery cleric Invoke Duplicity (1 minute)
• Archfey warlock Dark Delirium (1 minute)
• Draconic sorcerer Draconic Presence (1 minute)

This argument applies equally to all of them, and any other abilities that say "as if concentrating on a spell". This seems totally normal and intended, not a problem with this reading of RAW.