The basic structure of a formal letter: The rules might be arbitrary, but taken together they create a formula that produces clarity. When a letter follows all the rules, the reader doesn’t have to struggle to figure out where a piece of information might be, what action they should take next, or who sent it _ everything can be found in its place. Knowing the rules of letter writing is, in that sense, akin to knowing the rules of page layout or web design _ the format of the document should support and clarify the content.
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.