Business writing varies from the conversational style often found in email messages to the more formal, legalistic style found in contracts. A style between these two extremes is appropriate for the majority of memos, emails, and letters. Writing that is too formal can alienate readers, and an attempt to be overly casual may come across as insincere or unprofessional. In business writing, as in all writing, you must know your audience.
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.