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.
I can’t tell you how many truly awful letters I’ve had to read in my life. As a former administrative assistant, I was responsible for all the unsolicited submissions to a major New York museum. Artists and collectors all over the country (and abroad) wrote to have their work considered for display or acquisition, and to be honest the decision frequently rested more on the quality of their cover letter than on their work _ which, romantic dreams aside, rarely if ever sells itself.