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.
A Note About Email Business Letters. It might be tempting to leave out the closing when you're communicating by email, but don't give in to that temptation. Although no_closing emails are perfectly fine for everyday communication with your friends and teammates, they'll seem brusque – or worse, unprofessional – to people you don't know as well. You should also use a business letter closing when you're corresponding with someone professionally about an important issue, whether it's a new project or a job opportunity.