Just because you can't actually pay it doesn't meant they won't bill you for it. That's just going to generate a debt to Amazon once the money runs out. It won't magically turn off your AWS charges...unfortunately.
I’ve had the card I use for AWS expire in the past, so I’ve got the billing notifications from them. Basically they start emailing you as soon as the charge fails and give you a couple months before they shut off your services. As long as you pay the past due charges by then, there are no complications. Based on the wording in the emails, it seems like they will just shut down the account if you don’t pay the past due charges. Which feels like the right outcome to me. I suppose they could send you to collections but companies that do that tend to be more upfront about it.