Learn Arabic (Modern Standard): باب مصعد الطابق مغلق الآن.

In a complex Idafa (a chain of three or more nouns showing possession), only the very last noun can take ال (the). The earlier nouns (باب and مصعد) must never have ال, but they are still understood as definite because they are linked to the definite final noun الطابق (the floor).

If I read this aloud with all the vowel endings, how do the cases work in this three-noun chain?

The first word باب (door) is the main subject, so it takes the nominative ending damma (u). Every subsequent noun in an Idafa chain is the "owner" of the previous one and must take the genitive ending kasra (i). So you would pronounce the chain as: baabu mus'adi t-taabiqi.

Does the word مغلق (closed) describe the door, the elevator, or the floor?

It describes باب (door). In an Idafa chain, the first word is the core subject of the sentence. Because باب is masculine, the predicate (the information about the subject) مغلق is also masculine. If the first word had been feminine, the predicate would also need to be feminine.

Why doesn't مغلق have ال at the beginning if we are talking about a specific door?

مغلق is the predicate, which means it provides the "is closed" part of the sentence. In Arabic verbless sentences, the predicate is usually indefinite and drops the ال. If it were written as المغلق, it would act as a connected adjective meaning "The floor's closed elevator door..." and the sentence would feel incomplete.