No, your bank account will not be closed merely because you abandon your green card. In fact, your bank will have no way of knowing that you’ve done so unless you tell them. Note also that there is no law that prohibits a nonresident alien from having a US-based bank account.
However, if your account remains inactive for a period in excess of the time allowed by the law of the state in which your account was opened, the account may be closed and the contents handled as “lost property” according to the law of that state. You can reduce the likelihood of this by keeping your address current with the bank, and by responding promptly to any communications they send you regarding the account. In addition, many banks charge “maintenance fees” to inactive accounts that will cause those accounts to eventually reach a zero balance, at which the account will be automatically closed.
