I've heard that Diablo III requires a persistent internet connection even in solo play. Does anyone have any sources to this effect?
Yes. This is a necessary consequence of Diablo 3's online cash auction system. All your stuff has to be stored on the server to prevent 'fraud'.
As you can see, the official system requirements include a broadband connection.
On further investigation, there appears to have been a rumour at some point that Diablo III will feature an "offline" mode, but this information is a year old, and Blizzard's system requirements seem to contradict it.
Q: Is there going to be an open/closed battle.net like on Diablo 2?
A: Very little is known right now on how Battle.net is going to work with Diablo III, mainly because they haven't done much work with it yet. It is unknown right now whether or not Diablo III will have an open/close Battle.net option. There will be an off-line mode but it is unknown whether or not you will be able to use those characters to play on-ine (sic).
A more recent and presumably up to date FAQ on the Diablo III wiki appears to agree that there will be no offline play.
Right down the bottom is an unqualified "Internet connection required". There's no "for multiplayer games" or "for some game modes".
Just in case you're one of those people who are living in hope that Blizzard will relent, and add an offline mode just before (or just after) release, here's my best guess:
Diablo3 is online-only only in all forms, from the ground up. It's not an action-RPG that Blizzard has bolted an online-only requirement on. It's more like WoW stripped down to instances-only gameplay, where the loot generation, combat mechanics and probably even mob AI is all handled server-side.
Not only does this allow them to leverage their dev's experience with WoW as well as soak up surplus server capacity, it makes Diablo3 incredibly difficult to pirate. Blizzard is betting that those upsides beat the downside of annoying a good percentage of their customers. I suspect, sadly, that they'll win that bet.
I've no doubt that eventually someone will reverse-engineer a fake backend that will allow offline (and unlicensed) play that rivals the real thing, but it won't be any time soon.