"I wear a tie at work."
I understand that "wa" indicates that "at work" is the topic, but coild this sentence be written without it? If not, why is it necessary? If so, then how does the sentence differ with and without "wa"?
The は indeed refines the topic or scope of discussion, and doing that explicitly has the effect of implying that the same might not be true elsewhere.
Sometimes it helps to try translating 「Xは～」 as something like "As for X..." or even "If we're talking about X..." to try to simulate the effect that は has in English. So here we'd get "As for at work, I wear a tie." or slightly less stilted, "If we're talking about at work, I wear a tie."
Compare that to simply "I wear a tie at work." which places less emphasis on the context.
Most of the time in English we'd accomplish this emphasis just through intonation, but in a language which has this built in idea of context or scope it's a natural way to do it.
It is not necessary. The は here stresses that you wear a tie at work, but not anywhere else.
why not 'Shimasu'? I have never heard 'shimemasu' used in this context.
Japanese uses different verbs for wearing different parts of clothing, in the case of items that can be tied or fastened, they use 締める (しめる・shimeru) which is a verb that means "to tie, to fasten"
You can read more here: http://www.guidetojapanese.org/blog/2010/01/16/you-cant-just-wear-it/
I sometimes see する、つける used with ネクタイ so they are probably OK. However the best verb to use is しめる.
I would be less concerned by the insistence on using "de" instead of a stressed "wa" if the lesson hadn't already used this exact sentence with the "wa" for emphasis.