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 the end, this is all about respect for your recipient. If you’re offering a real solution to a real problem, people want to know about it. By following proper and well_understood formats and keeping your writing clear and engaging, you’re making it easier for them to solve their problem. Provided that you’re honest and have the skills and talents to back up your claims, writing a good, solid letter makes it more likely that both you and your reader will be satisfied. Try it the next time you have to write an important letter.