Not only does this mean that you should focus on neutrality in your letter’s appearance (avoiding the use of colorful paper, brash logo designs, and artistic fonts), but it also means that you should employ a very low_key, unnuanced, and professional closing phrase. The best_case scenario is that the hiring manager, colleague, or connection won't even notice the closing.
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.