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.
In these days of email and instant messaging, letter writing is becoming a lost art, and especially the formal letter. I’m pretty sure that formal letter rules are still taught in schools, but very little of it sticks. If you’re an executive or other high_ranking person within an organization, you can probably rely on your secretary or assistant to handle the niceties of letter writing, but for everyone else, knowing how to write a strong letter ourselves is important.