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.
Reread the description of your task (for example, the advertisement of a job opening, instructions for a proposal submission, or assignment prompt for a course). Think about your purpose and what requirements are mentioned or implied in the description of the task. List these requirements. This list can serve as an outline to govern your writing and help you stay focused, so try to make it thorough. Next, identify qualifications, attributes, objectives, or answers that match the requirements you have just listed.