Q: My elderly father with dementia forgets to pick up the mail. His mailbox is overflowing, bills are overdue and unpaid. What should I do?
A: I suggest that you engage a daily money manager (see http://www.aadmm.com/) who is bonded and insured to help your aging Dad.
A daily money manager will arrange to pick up the bills or transfer them to a different address and will then pay the bills, reconcile the bank statements, and take care of other related paperwork such as getting things together for the tax preparer and dealing with medical insurance issues.
In some cases, I will sit and do the bills with my client, while in other cases, they (or their children) prefer that I take over the job. Which way you go will depend a lot on your Dad's cognitive status. You mention that he has dementia, and it is not unusual for paranoia to accompany it, creating a situation where it is better for the daily money manager to do her work at arm's length.
On the other hand, if I have a client who is in great shape cognition-wise but suffers from vision problems, I'm much more inclined to sit and do the bills together with her to help her preserve her independence and dignity.