iPhone Mail.app - Showing Disabled Email Accounts
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.