Some newspapers are completely free. Most charge a tiny amount to avoid the stigma of being a free paper (or, gain the credibility of a for-pay paper). The majority of the paper's revenue comes from advertising -- which also works online (setting aside the ad-block arms race).
As for investigative journalism, I suppose it's just a matter of finding someone willing to pay for it. There's the BBC approach of having everyone pay a little bit for relatively NPOV news, since that's worth a little bit to everyone. Alternately, there's what this article discusses: many small groups are willing to pay more for investigation that may support their own agendas -- or do the investigation themselves, and blog it.
On the Internet, the cost of the delivery (bandwidth and server costs) is too low on a per user basis to bother charging, so it's "free" but the paper is free to make money the same it does with regular newspapers - advertising.
If anything kills the concept of newspapers, whether online or not, it'd be the self-righteous jerks who run ad blockers. Though, of course, this would lead to in-line ads, and an ever blurry line between editorial and advertising as ads get stuffed into the editorial.