I found this thread thismorning. This has been going on with my S4 since day one. I am not shore how to fix it but I know what is happening thanks to my wife.
My phone is setup to recieve the same emails that get sent to my Outlook account on my laptop. Thismorning we sent myself an email which apeared on my phone, opened it, refreshed it, closed it exited out ect. Nothing we could do would make it go away.
Then! We turned on my laptop, opened outlook and it automatically brought in my email. The next time I checked my phone, the email was gone.
Did it over and over. Same result.
So your email remains on your phone up until Outlook brings your email in and then the next time your phone is refreshed your email will be gone from your phone.
I have no idea what to do about this? But am wondering if the people that are having this problem are all using Outlook on there PC? If so then its probably an Outlook problem. If we are using multiple email software packages then its a problem with the S4.
Jeff