The Mezzanine

I almost never write letters, lol, so if I do, it's probably a formal request, so I'd use "Dear Mr. Smith." (Unless it's someone I've met and call by their first name in person, then I'd say "Dear John," but I can't imagine a reason to write a formal letter to someone I know on a first-name basis.)

With email, I'll generally use the first name, whether I know them or not. The only exception is if it's an elected official or otherwise important person, I'll usually start with "Mr. Smith" and then switch to "John" if it's appropriate and I feel comfortable.

I also tend to address my professors as "Dr. Smith" or "Professor Smith" in email unless they've said to call them by their first name.

ETA: And as others said above, I almost always use "Hi" instead of "Dear" in emails.