There have been many reports of email trouble with iOS 10 that even I have experienced with my iPhone 7 Plus.
You may get an error message that says Mail cannot move or delete an email. It may look like the email was moved or deleted, but if you exit out of the Mail app and then go back in, you will find the offending email has returned to your Inbox.
To fix this, go into Settings > Mail. Select the account that you are having trouble with. You will see several options that you can use with the account including Mail, Contacts, Notes, etc. Mail will be toggled ON. Toggle Mail to OFF. The switch will go from green to white, indicating it is off. Wait 60 seconds. Toggle it back on. Double tap the home button, which will show recently used apps. Swipe up on the Mail app to force quit. Then go back into your Mail app.
I only encountered this problem once, and the above fixed it. Some people have reported that this has happened to them several times. It’s a known issue, and I hope we see a resolution in future updates.