How do you know for sure whether or not to use a closing? One good test is to ask yourself whether this email is more akin to an instant message/text or a business letter. If you're giving your teammate a quick update on an ongoing project, a formal closing might not be necessary; if you're throwing your hat in the ring for a promotion, it definitely is required. When all else fails, and you're still not sure, err on the side of caution and include it. You'll never go wrong by being too polite and respectful.
Writing for a business audience is usually quite different than writing in the humanities, social sciences, or other academic disciplines. Business writing strives to be crisp and succinct rather than evocative or creative; it stresses specificity and accuracy. This distinction does not make business writing superior or inferior to other styles. Rather, it reflects the unique purpose and considerations involved when writing in a business context.