In German you most definitely do.
I'm not sure about English, but I think you do, too.

Why don't you write (ZIP codes in AE) or (ZIP codes [AE]) instead?

Thanks, Penguin. I've decided to write is as you suggested- but I'd still be interested to know what the rule is.

Brackets within brackets
Brackets within brackets look complicated. It may be better to avoid this by wording things differently.
If you do use brackets inside brackets they need to look different. You could use square brackets inside rounded brackets.

Example
There are several different ways of beginning to write an essay (all of them equally helpful [Drew and Bingham, 2001]) and it depends on writers which they prefer.
http://universitywriting.shu.ac.uk/punct/advi...



Some authors follow the convention in mathematical equations that, when parentheses have one level of nesting, the inner pair are parentheses and the outer pair are square brackets.
http://en.wikipedia.org/wiki/Bracket

Couldn't you avoid the problem and write "postcodes / zip codes" or "postcodes (zip codes)"? Presumably AE readers will have the hit to work out which wording applies to them. Or is the text going to be adapted, depending on whether it is intended for a BE or AE readership, with the bit in brackets being a note for the editor?

Incidentally, I can't see anything wrong with using nested brackets anyway.

The general rule that was asked for in #2 is simply: Close any bracket (parenthesis) that you have opened. If you have openend 2 parentheses, then you must close 2 parentheses.
BTW:
I have been told that () are "parentheses", and [] are "brackets".

Hier ginge doch auch:

(ZIP-codes, AE)

oder

(ZIP-codes -AE-)

Es sieht zwar komisch aus, wird aber in D so gemacht (das Klammernschließen nämlich (also innerhalb einer anderen Klammer)).

:-)

Sieht es denn komisch aus? Erst recht komisch würde es doch aussehen, wenn die zweite schließende Klammer fehlen würde, bzw. der Leser würde davon ausgehen, dass diese weiter hinten im Text steht. Das gilt im Englischen genauso.

I think the second set of parentheses is truly unnecessary, and would go with judex's first suggestion
(ZIP-codes, AE)
--but without the hyphen:
(ZIP codes, AE)
Or perhaps this:
(or ZIP codes, in the US)
It depends in part on your audience: will they all be familiar with the term "AE"? And besides, "AE" indicates it's a linguistic difference, whereas I suspect it's a difference between postal systems. But I could be wrong. I only know ZIP stands for Zone Improvement Plan, a system that was introduced in the US some 50 years ago, more or less.

PS: I guess I neglected to answer the actual question: Yes, it is important to close all the parentheses that have been opened, no matter how it looks. Mixing it (i.e., parentheses and square brackets) might be a good option in many cases, if you don't like the look of ))
(my double-chnned smile) :-))

I'd close the parentheses and delete the hyphen between zip and code. You could then write "(AE zip codes)

#10: + i = chinned (whoops)