No, a linen closet is a linen cupboard.
To qualify as an airing cupboard there needs to be a provision for air circulation, usually heated. Une armoire à linge does not have this, it is for storage not airing.
According to my dictionary (OED) the French for airing cupboard is "placard qui contient la chaudière et où l'on range le linge", in other words, there is no French name for it.
An armoire à linge is a linen cupboard.
What's a linen cupboard? (please describe without recourse to the terms "cabinet" or "closet" owing to the confusion laid out below ;)
The only modern definition of "cupboard" here is "cabinet or closet or other piece of furniture with shelves..." Meanwhile "cabinet" is defined as "1. storage closet either separate from or built into a wall" or "2. cupboard": https://en.wiktionary.org/wiki/cabinet and "closet" is "a private cabinet" (whatever that means): https://en.wiktionary.org/wiki/closet#Noun
Oxford, incidentally, explicitly lists "wardrobe" as a kind of "closet": https://en.oxforddictionaries.com/definition/closet
Out of this morass I feel that my presumptive personal predilection to call anything where you store linens a linen closet (out of lack of personal familiarity with any other term) has at least some justification.
A picture is worth 1000 words, and here are 1000 pictures:
This should settle most arguments!
Before this, I had never heard of "armoire". Linen cupboard or wardrobe-yes. I put wardrobe and marked wrong. Grrrr.
Looks like there is: http://context.reverso.net/translation/french-english/placard+%C3%A0+linge
But it's a comparatively less common term: https://goo.gl/bkabhA, whether due to reasons only of language or of architecture I don't know.
EDIT: I got industrious and looked into the architectural difference question. This is mostly about Switzerland, but gets at the point that built-in closets are much less common on that side of the Atlantic: https://www.englishforum.ch/housing-general/57844-no-closets-where-do-you-hang-clothes.html
One comment interestingly notes that the non-built-ins commonly found are unusually large by e.g. U.S. standards for non-built-ins.
I don't know what either of those things are, but if the translation info I find for them are anything close to right, I don't know that they're referring to the right sort of item.
I assume a "hot press" is something that heats up? Its translations are quite different: http://context.reverso.net/translation/english-french/hot+press
If Larousse knows what it's talking about, I wouldn't be too sure about "airing cupboard," either: http://www.larousse.fr/dictionnaires/anglais-francais/airing_cupboard/561703
My answers are
"here is a big laundry cabinet"
"here is a big laundry closet "
Both are rejected buy duo. Is "laundry cabinet" or "laundry closet" wrong?
Must we use "linen"?
---just a bit off the topic, as per my life, I have never seen any closets using the material linen. Have anyone seen that?
I'm unfamiliar with either "laundry cabinet" or "laundry closet." The former sounds like it might be some sort of built-in hamper, sort of the opposite of a linen closet (which is for clean linens).
A linen closet isn't a closet made of linen. It's a closet that holds linens. Consider "soccer field"; it's not a field made of soccer ;)
In American English we have adopted the word armoire from the French. It's a piece of furniture, which I don't think of as ugly as the pictures Google presented, but I digress. It is used as a free-standing closet which you could put a number of different types items. So, I believe that linen armoire should be accepted.
It's simply cultural differences which Duolingo does not understand and will only choose one or two translations to mark as correct. Of course, colloquial expressions abound and we have all been brought up with different ones. I suppose there isn't one correct answer but it's a pity Duolingo does not have a better wa to work this out
Such matters are worked out via sentence reports. Duolingo's contributors can't be expected to come up with every valid translation for every dialect of English on their own.
Those wondering why their translation was not accepted should review the matter and if it indeed proves to be a valid translation of the French sentence, report it as missing, or ask about it if uncertain.