I have been having the same problem since last Thursday. Spent nearly 2 hours on Tech Support with a 2nd level technician. I was at home with 2 iPhones, 2 iPads and a couple computers available to be used while we investigated the issue. It is a very strange behavior. Here is a summary of the results of our efforts.
1 - If you mark AOL mail as read on an Mac using Mavericks or Snow Leopard, it will be marked as read on iOS 7 on your iPhone or iPad within seconds.
2 - If you open an AOL e-mail on your iPhone - by touching/loading it - it will stay marked as read on your iPhone, iPad and on your computer.
3 - But, if you use the EDIT function - and try to mark an email as read (or mark all as read using the command at the bottom) the iPhone will initially mark the emails as read but they will NOT be marked as read on the AOL server. Your computer and any other iOS 7 devices will not change the e-mail to read. At some point, typically within 5 minutes, the message will reappear on the iPhone as unread.
It seems to be tied to the EDIT function in iOS 7 mail as Read, Flag, Move, and Delete all fail to work in the same fashion. It is also a one way error - iPhone iOS 7 to the AOL IMAP server - but only if you take action without viewing/loading the e-mail.
Note that we tried multiple mail account deletes, various settings deletes, and even a total iPhone reset. Nothing fixed this error.
AOL mail on my iPad, using iOS 6, works as it should. AOL mail on my wife's iPad, using iOS 7, has the exact same problem as my iPhone 5s.
The 2nd level technician has concluded it is an iOS 7 bug and forwarded the information to the engineering team.
If others are having this issue too - please add it to this thread. We need to make sure Apple understands it is a wide spread issue that needs attention.
Tim