I think only the initial emails need the greeting, or if its been more than a day. If you are replying back and forth and the messages are short (so, not really a letter, more of a conversation) the greeting would be redundant. The same goes for the closing, writing sincerely every time would be over-kill in a series of short emails.
Thank them at the end for answering or for their time if you're waiting for a reply, tell them you're welcome if they've thanked you, etc etc, and possibly sign the end to help keep them from being confused about who the message is from if the email service you use doesn't already do that for you or display your name next to your email.
While it's impossible to be too polite, you won't seem unprofessional if you drop your greetings and more formal closings in short exchanges.