Diablo 3, here's what happens. First, your game will lock up for a minute or so, then you'll be sent back to the character selection screen.
According to Eurogamer, at this point you'll log back in and find that you've retained all gold, experience and items – no penalties there. Once you get back into the game, you'll start from your last checkpoint.
The decision to require a constant internet connection initially received some blowback from the community, but it would seem it's not as bad as initially thought, at least in one regard. Eurogamer even notes that being booted back to the character selection screen didn't happen every time – occasionally the game would resume as if no disconnect even occurred.