"I wear a tie at work."
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.
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/
Good question, you can actually say 仕事はネクタイを締めます。 It is called こんにゃく文 (Konjac sentence).
However Duolingo wants you to understand in a normal case, the particle で cannot be omitted because it is to represent the location of an action. Whether you can omit で depends on context.
Closing a door or window implies (or at least suggests) fastening it by a latch to its frame, and putting on a tie implies fastening it with a knot round your neck. Same again with a belt -- there is an act of fastening. One difference: if you actually lock the door or window, the kanji is the same (締める), but if you just push it shut without locking it, the kanji is different, but the pronunciation stays the same (閉める).
「仕事でネクタイをする・します。」should be accepted.
Dictionary of Basic Japanese Grammar
Do; make; play; play the role of ~; wear
Kyoko is wearing a beautiful scarf.
You’re wearing a nice tie, aren’t you?
The Direct Objects of Key Sentence (F) and Example (f) are items that cover a small part of the human body such as ネクタイ '(neck)tie', 手袋 'gloves' and 腕時計 'wristwatch'. When used with such direct objects, する means 'wear'.