If your intent is to persuade the recipient in some way, whether it's to invest money, give you a reference, hire you, partner with you, or fix an issue, create a compelling case for your cause. If, for example, you want the reader to sponsor a charity event, identify any overlap with their company’s philanthropic goals. Convince the reader that helping you would be mutually beneficial, and you will increase your chances of winning their support. Keep your closing paragraph to two sentences. Simply reiterate your reason for writing and thank the reader for considering your request.
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.