In the U.S.? Nothing. Sorta. A privacy policy is a good idea and helps trust organizations such as eTrust evaluate your site for trust. It also helps the site user. I always read the privacy policy when any account or PII (personally identifiable information) is taken.
The exception is where a site engages in marketing and serving to children 13 and under. There are legalities involved and a privacy policy is required. http://en.wikipedia.org/wiki/Children%27s_Online_Privacy_Protection_Act
If your site is clearly not for children, then it is just good business even for sites that do not have account registrations but do capture usage information, uses cookies, or uses a 3rd party tool that may do the same such as Google Analytics and Adsence.
If your site is purely informational and you do nothing else but evaluate site logs, then I would not worry about it unless you want a trust organization to give a better score.
The Wikipedia page on this is very good. http://en.wikipedia.org/wiki/Privacy_policy Here you will see as much information as there probably is with good links on the subject.