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.
Formal communication is on the wane in modern life, but there are still times when it's the only correct way to reinforce a connection or convey information. If you're applying for a job, looking for a recommendation, or expanding your network, err on the side of formality. Don't let the slightly archaic feel of a formal business letter tempt you into using flowery, outdated language.