How to Empty the IMAP trash folder I have an iPhone 5s with iOS 8. There are about 80,000 emails in an IMAP trash folder. How do I get rid of these emails? 

Perhaps my phone was set up incorrectly but I have no reason to want to keep these.
The setting that you are looking for is buried in advanced mail settings.

Open Settings > Mail, Contacts, Calenders > (Choose the IMAP account in question) > (choose it again) > Advanced 

This screen will have a section titled "Deleted Messages", Set "Remove" to an option other than "never".

For what it's worth, it's the mail client, not the mail server, that is responsible for cleaning up the trash folder.  The client is the program or app that you use to read email, in this case your iPhone.  Many people have problems with this setting when using multiple clients, such as an iPhone, iPad, and Mac or PC at the same time.  In that case if the clients don't all have the same thing for this setting, whichever has the shortest time period is the device that will prevail.

Note: changing the setting should fix your issue, although it might not be an instant fix. I'm not positive how it will behave with 80,000 trash emails to cleanup.

Alternate: A more immediate method appears to be, open mail which will open your inbox, back up one screen by clicking whatever is the the top left (likely the account name, it will vary depending on how your phones is setup), then in the Accounts section choose your "Account" (not your Inbox for that account), from there select the trash folder, click "Edit" on the top right, and then "delete all" in the bottom right,  and "delete all" again to confirm.  Also be sure to adjust the advanced setting above so that in the future your trash folder will be maintained automatically for you.