iPhone Mail.app - Showing Disabled Email Accounts I have lots of emails on my iPhone from an old account that I no longer use. Mail reminds me often that it cannot connect to the account.

When I uncheck the account from email preferences, the  account disappears completely from Mail.

How can I keep the emails in my application so I can browse through them, while configuring Mail not to try checking for new mail?

iPhone 5s iOS 7.1.2
When you remove an email account from the iOS mail app, it removes all the emails (and other data) from the device. There is no way to keep mail on the device from an account that was removed. 

If you leave the account(s) signed in to the device, you can set the mail app to only fetch new mail manually. This would allow you to have the mail on your device, without getting new mail automatically.

EDIT

To fetch manually, go into Settings > Mail, Contacts, and Calendars > Fetch New Data. There you will see a section at the bottom called Fetch and set it to manually.