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.
Remember, you're hoping the person who receives your letter has no memory of your closing at all. The last thing you want is a hiring manager going into an HR meeting with your cover letter in hand, asking the team if they want to meet with "Mr. Kindest Personal Regards."