This is actually something that might just happen in Norway. Some politicians want to get rid of paper money by 2025. Currently the law says that every service or store are required to accept cash as a payment method.
In order to shift over from paper money, everyone will need a bank account, even small kids who want to buy an ice cream.
If we all go away from paper money, the banks will have full control over the economics in the country. Unless the country starts accepting bitcoin as a payment method, I think that this is a very bad idea.
I don't think this is a very likely event
In fact, there are a lot of reasons to diminish the use of paper money (and cash as such), but they have mostly nothing to do with governments wanting more control over the finances of their citizens. If there is no cash around, bribes and illegal payments will still be made with gold bars and coins (or even with bitcoins, wtf). Cash won't go away completely but its use is likely to get limited through pure economic means. In other words, where cash is more handy, people will continue to use cash. If cash is outlawed directly, people will just start using some other money token (so-called monetary surrogate) instead of official currency, so there is no sense in outlawing it completely