None..... bank accounts use chex for history reporting. Revolving accounts, etc. use the three CRA's (Equifax, Transunion, Experian).
The only negative this can have is if you close, say, a Bank of America bank account, then reopen the Bank of America bank account. Banks like these tend to pull "hard inquiries" on your Credit Report when opening an account.
Also, internally, a company like American Express may conduct a Financial Review (FR) if you have revolving credit and/or a charge card with them. If you tell them you no longer have a bank account, it may cause concern...