Earlier this month I saw a question on Money.SE asking whether it was possible for a US non-resident to open a bank account? I thought that question was confusing as it didn't specify whether the US non-resident alien was living in the United States or not. A more interesting question I think would be the following:
Non-resident aliens living in the United States are able to open bank accounts. Suppose a non-resident alien living in the United States falls out [immigration] status. What happens to the bank account and to the money in the bank account?
Nothing happens. A bank is a business; your relationship with the bank doesn't change because your visa or immigration status changes. Money held in the account is still held in the account. Interest paid on the account is still taxable. And so on.
If the account is inactive long enough, abandoned account rules may apply, but that still has nothing to do with your status.