When you write a business document, you must assume that your audience has limited time in which to read it and is likely to skim. Your readers have an interest in what you say insofar as it affects their working world. They want to know the “bottom line”: the point you are making about a situation or problem and how they should respond.
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.