I understand that the buyer needs to update their billing information and will get an e-mail from PayPal saying it is going to expire and asking the user to update their billing information, but, what happens if the buyer only have one funding source and does not update the billing information? The payment could not be processed, but in theory, the billing agreement does not change its status from "Active" to "Suspended", does it?
That would depend on the type of billing agreement.
If it's a recurring profile that automatically bills, it will eventually become suspended due to lack of payment. There may be retries before the suspension, depending on how the profile is set up.
If it's of type "merchant initiated billing" for at-will reference transactions, then it remains active.