If all of your apps save directly to Dropbox your 'hard-drive' and thumb-drive are no longer important, and in that way they are 'replacing' the hard-drive, even if not in technical terms, the sentiment is correct.
That's not possible, since your local storage will always be faster than a networked one.
Network storage has always been slower than local, and until we get 10Gb to the home, will continue to be so.
For everybody's most important app, the browser, local storage is already basically irrelevant. That's trained people to be perfectly fine with little network delays. Things like Google Apps show that data and apps can easily live elsewhere and it's basically fine.
In the long term, I think the only question is whether data+apps will live together with the app provider, like Google Docs, or whether data storage will stay generally separate, as DropBox would surely prefer.
There will still be some of us who keep big local drives because we really want to know where our files are and control their movements. But for everybody else, local storage will just be cache. Which means whether it's spinning rust vs flash vs RAM is an optimization detail. And that decision will mainly be made by hardware product designers, not users.
An interesting thing to think about in this context is how the long-term costs of storage are falling. In the last 20 years, RAM is ~4 orders of magnitude cheaper:
http://www.jcmit.com/mem2013.htm
Flash is an order of magnitude still cheaper for now. But if networks keep getting better and services like Dropbox get more solid, that may not matter. If you assume trends continue, then putting half a terabyte of RAM in a device may be cheap enough that the added complexity of flash just isn't worth it.
Yeah - this question intrigues me. I wonder how Dropbox intend to expose all the useful "filesystem meta data" to all the relevant OSs? How is Spotlight on my Mac going to work if local storage is not cannonical? How am I going to grep an entire "local" git repo if the files are mostly not local?
And I wonder if Dropbox even care about those things? Are they both indicative enough of a "power-user" and that they know the "mass market" isn't ever going to ask those questions, when presented with a "home-machine-less Dropbox" which they can access mainly on their phone/tablet, and occasionally on a shared/borrowed/rented full sized laptop/desktop machine?
You don't always (perhaps even "often") _need_ the raw speed available from locally attached storage. Sure, I'm not going to do video editing or keep my Database indexes or Photoshop scratch disk on internet-connected-storage. But there's lots of files in my homedir that could easily be relegated to "it might take a minute or two to retrieve this file's contents" grade storage…
And Dropbox _already_ has a ton of data on how people are using Dropbox on iPads and similar devices. My iPad _pretends_ to have my entire Dropbox folder locally, but doesn't really. Sometimes that means I need to wait a minute or two while a document downloads over 3G to my iPad while I'm in a café - if that means I get to spend 18 minutes working on that document instead of the whole 20mins I've got available, that's a much bigger win than me spending zero of those 20 minutes working on it.
Microsoft tried that assumption with the Xbox One, and we know how that went. :P