Can the Armory district card destroy a district in a completed city?
Apparently Bruno Faidutti (the game designer) intended to allow this.
The Warlord has the following limitation (as found in the official Citadels rules):

You may not, however, destroy a district in a city that is already completed by having eight districts (or seven districts when the Bell Tower is in play).

But apparently Faidutti intended for the Armory not to have this limitation.
As found on BoardGameGeek, the Warlord would even be able to destroy a district from a city that once was completed, but became uncompleted because of the Armory:

Nonetheless, here is how, according to Bruno Faidutti's ruling, a the final round may play out:
Assume Ashley (a Magician) should build an eighth district first, triggering the end of game condition.
Brad (the Bishop) pops his Armory to destroy Ashley's Cathedral (her only religious district). Even though she has seven districts, the game will still end when the round finishes.
Clara (using the Architect) builds twice, bringing her district count to nine.
Finally, David (acting as Warlord) wants to destroy a district (the gold is useless to him otherwise). David cannot destroy anything of Clara's, since her city is complete, and Brad is the Bishop, so he is safe. Ashley's is no longer complete, so David destroys another of Ashley's districts.

There is some ambiguity as to whether the game would still end after such a round (imagine that Clara in the previous example would not have completed her city) and as to which bonuses Ashley in the previous example would receive.