"You are not allowed to use a dictionary during the test."
Most frequently you use 何何 では。。。 ありません
How I understand it after Googling is that using only で...ありません sounds kind of odd in Japanese. Similar to when non-native English speakers say something understandable but grammar wise it's just a bit off and not how native speakers would phrase it.
ではありません。(ではない) Means "(subject) is not"
In the right situation you could instead use: でもありません。(でもない) which means "(subject) is also not"
テストでは In the test (literally, as for what happens during the test) てすとは The test (literally, as for the test)
So in the first example, you're saying DURING THE TEST (focus of the sentence) (stuff will happen/not happen/is not allowed/is happening). In the second you're saying THE TEST (focus of the sentence) is (hard, easy etc).
I'm a novice in Japanese but I think, pertaining to English, the verb "引く" applies the the use of a dictionary and not necessarily "use" in a generic sense. Just as much as individual verbs fir individual action would have the same applications. Nevertheless, it should be accepted.
The double use of は doesn't really sit well with me for this one. Is it normal?
I would have thought that 「テストでじしょをひいては、いけません」("As for using a dictionary during the test, you may not") would be more natural, right? As opposed to what is suggested 「テストでは、じしょをひいては、いけません」, which looks like it's equivalent to "As for during the test, as for using a dictionary, you may not". I just added the commas in the above quotes to clarify my thinking. Not sure if their usage is correct.
Is it because では are being used together and it changes how the particles work a little?
First off, it's perfectly reasonable to have multiple topics in a sentence, and thus have more than use of the は particle, so you're definitely wrong there.
Second, you're parsing the sentence wrong. It's not 辞書を引いては、いけません, but 辞書を引いてはいけません. 〜てはいけない (〜ては行けない) is a construct used to express prohibition.
Examples at https://context.reverso.net/translation/japanese-english/%E8%BE%9E%E6%9B%B8%E3%82%92%E4%BD%BF%E3%81%A3%E3%81%A6 would suggest otherwise.
So the first では is to say DURING the test. And the second one is more for marking the subject which is the use (or the act of opening) the dictionary.
So to break it down: テストでは(during the test)じしょをひいては(the act of using or opening the dictionary)いけません(must not do/or not allowed to).
(Please correct me if im wrong)
"テストでは辞書を使ってはいけません" still isn't accepted after 2 years. But it's an exact literal translation of the given English, and in exactly the same format as the accepted answer, just using the verb 使う vs 引く。 Is the verb 使う never used with 辞書? (A google search suggests it's actually quite common, and there are lots of examples at https://context.reverso.net/translation/japanese-english/%E8%BE%9E%E6%9B%B8%E3%82%92%E4%BD%BF%E3%81%A3%E3%81%A6)
You don't need a particle, but using に should be acceptable (not で. テスト中で would refer to something located inside the test, and be interpreted as テスト(の)なか instead of テストちゅう).
I would suggest, though, adding a comma after 中 if not using a particle.
Alternatively, you can also say 「テスト期間中。。。」
Grammatically it should be fine. By placing 辞書 at the beginning of the sentence you might be slightly overemphasizing it as opposed to the テスト. This structure would make more sense if you were constructing a sentence that said something like 'While you can't use a dictionary during the test, it is fine to use it during regular lessons', or something like that. 辞書をテストではひいてはいけないが、一般授業ではひいても問題ありません。
テストは辞書を引いてはいけません。Is accepted, I think the problem anyone may be having is that で alone is unacceptable because the test needs to be defined as the topic at the very least.
Especially in these longer sentences it is not a good idea to continue without defining a topic or at least a subject so simply で is just unacceptable. は alone is probably a little general for the nuance of the sentence, however at least you defined which noun is the topic of the sentence.
Hope that was correct or at least helpful to those who are having a tough time with it.