I think the main issue with that is more a matter of human nature than anything else. Even with a good-faith agreement, working in a format that has 24-hour phases can make it very easy for people to forget they're supposed to be moving along faster. For example, having to refresh the page manually to see new developments, chats, etc. What we'd want is the live format to naturally encourage live behavior, such as having moves/chats auto-refresh as they occur (which isn't nearly as hard as it sounds from a scripting standpoint). The timer could be set by the host, but would default to, oh say 15 minutes, and cannot exceed something like 30 minutes and cannot be less than 3 minutes. And if a person logs-off (intentionally or otherwise), it gives the person 5 minutes (or until end of phase, whichever comes first) to return before their turn is automatically finalized as-is and the country goes into civil disorder. Of course this is just a conceptual outline; the specific details/numbers could be dealt with and discussed as the code is being written.
Regarding timezones, while your explanation has merit, keep in mind that most internet games are live, played by people across all different timezones, and it works just fine. Because an entire game would only last a few hours on average, anyone who's online when the game is first hosted will just join on the fly, typically already prepared to play it out for at least a few hours. I suppose another feature idea would be to allow the host to set a maximum game time, at which point the winner (if there's more than one player left) would be automatically determined by number of supply centers, or it'd just come out as a draw, etc.
Civilization IV is a good example of this (at least from a practical standpoint; the network code itself is a piece of garbage). A typical multiplayer game can go on for like 6 hours with roughly 7 players. If a person leaves, someone else can just join and take over that civ, just like you can now in phpDip. Of course, there's also others like Yahoo! Games, anything that uses GameSpy, etc. From what I can tell, this site definitely has a large enough returning player base to support a live gaming environment, since you're very likely to find enough people in or near your timezone to play a live game.
I think it'd be successful if the coding is done right. Certainly more effective than posting on a forum and hoping everyone remembers to finalize their orders quickly, since all it takes is one person to make the plan fail as it is right now.