Why would they need to move money that is denominated in Euros?
The two main fears would be loss of the money in the case of a bank failure and loss of purchasing power through a forced currency conversion.
In the first scenario, bank deposits are guaranteed by national authorities in the EU, but if your national government is broke and starved of Euros, how will you be made whole again if multiple banks fail? Even if you're a die hard supporter of the Euro, your money is gone. So to avoid that loss, you withdraw your Euros from Greek banks and keep them in the mattress or else deposit those Euros in a bank elsewhere, say, Germany, where you feel it is safer. This lack of a central FDIC-type institution is seen as one of the weaknesses of the current EU structure because bank deposits may migrate based on real or perceived weakness in one of the member nations.
In the second scenario, national authorities certainly could forcibly convert any bank deposits into a new currency and is one of main reasons for starting this thread--to observe what actually happens. One day, you discover your bank is closed, the ATM network is shut down, your debit card won't work anywhere and you can't check your online banking account. It's the cruelly festive sounding "bank holiday". Over the next few days or week, your one thousand Euro bank balance could suddenly be converted to one thousand New Drachmas and the banks and ATM machines stocked with new currency.
You can kick and scream all you want but the ATM or bank teller is only going to give you the new currency if you try to withdraw your money at that point. Various capital controls will almost certainly be put in place, such as making it illegal to carry Euros out of Greece (although the authorities will gladly welcome tourists bringing in Euros from outside Greece). The new currency will be worth less than the Euro, so suddenly you have significantly less purchasing power.
It's certainly not an easy thing to do and Greece will break all sorts of contracts and treaties and become an international pariah for a while, but it has happened many times, mostly in places like South America and Africa. It's possible that the government will renegotiate certain obligations and continue repaying them in Euros, primarily from its tourist industry, I imagine. No one knows what the details might be. And let's not forget that all these European countries converted
to the Euro not that long ago, so it's not that impractical a thing to contemplate.