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.
Consider the daily life of whomever you’re sending a letter to _how many other things might they have on their mind when they read your letter. Maybe they have a meeting in five minutes. Maybe yours is one of 200 letters they have to wade through. Maybe it’s the end of the day and they’re worn out. Giving them a concrete action to take makes it all the more likely that it will “stick” _ they’ll add it to their todo list or their calendar, they’ll pick up the phone, they’ll check out your website, or they’ll respond.