They can close it if you overdraw it too much, which tends to happen when you're poor. Hence people remain unbanked. Not to mention being homeless and having no where to receive mail. They may not refuse but where are they gonna mail your cards?
The format is:
John Q. Smith
General Delivery
Fort Wayne, IA 46806
Furthermore, there are free and reduce rate PO Boxes available in limited circumstances:
https://faq.usps.com/s/article/Is-there-mail-service-for-the...
And, for cases where "PO Boxes aren't accepted", there is a new format street address of the post office works with a PMB or # second line.