For uploaded or mailed cover letters, you do not need to include as much information as you would in an email message, because the heading of your cover letter includes your contact information. For a paper cover letter, use a formal business_style letter format which includes a heading, salutation, the body of the letter, closing, and your signature. Review these guidelines for what to include in your letter.
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.