I intend to apply for a postdoc position. The contact mentioned in the job advertisement, to whom I should address the cover letter, is someone I have personally met and talked to on a first-name basis (let's call him Dr John Doe). Therefore the natural form of address, for example in an e-mail, would be "Dear John". However, the cover letter for an job application is a formal document, so perhaps I should instead address him as "Dear Dr John Doe" or "Dear Dr Doe"? Thanks in advance!
I would lean toward the formal here, though it would be fine to remind the recipient of your past association, assuming it is professional and not personal. I doubt, however, that it is hugely important in most cases. But I also expect that you would already know if one or the other were greatly to be preferred.
In follow up contact you might change that, depending on the response to your initial inquiry. If he replies back as John, then John it is. But your letter will be seen by several others, as noted by user Anyon in comments.