The Apple Mail app doesn't provide you with the option to setup an out of office (OOO) reply - however, you can do it through the Mail app on your Mac. If you happen to use the more favourable and powerful Gmail app on iOS 10, you're in luck - as you can setup an OOO reply directly from the app. Here's how. Also see: iOS 10 release date
How to set up out of office in iOS 10: Out of office reply in Gmail
The Gmail app is free to download and works with a whole host of different email exchanges. As aforementioned, you won't be able to setup an OOO in the Apple Mail app.
Within the Gmail app tap the three white lines at the top left-hand side and scroll down to Settings. Tap on it and click on the account you want to setup an OOO reply.
Tap on Out of Office AutoReply and toggle the option on. You will now be presented with a variety of options, including the time you'll be away, the Subject and Message you wish to include in your automatic reply.
You will also be able to set it to send to your contacts only or your managed workgroup.
There are other apps out there which will allow you to synchronise your mail - such as Yahoo Mail or Outlook. Also see: How to use iOS 10
How to set up out of office in iOS 10: Out of office reply in Apple Mail
If you really wish to use OOO in Apple Mail, you'll have to use your Mac or iCloud.com.
Setup an out of office reply in Apple Mail on macOS
Note: For this section - you will need to have you Mac on, and with your Mail client open in order for it to work - as the Rules you set on your Mac are client side, not server side.
If you've got a Mac, you can set an OOO reply on the Mail program, which will then work if you use the Mail app on iOS.
On your Mac, click on Mail > Preferences > Rules > Add Rule - you'll now be able to see multiple options allows you to create an OOO for a set time period.
Once you've created your Rule - you'll be prompted if to Apply or Don't Apply the OOO reply to your selected mailboxes. Make sure you click on Don't Apply - or else it will send an email to all your contacts. Of course, if any new messages come in, that will work through both options.
Setup an out of office reply in Apple Mail on iCloud
If you do not own a Mac computer or prefer using the browser Mail app, you can access it through your browser.
Once logged into your iCloud, click on Mail > Settings > Preferences > Vacation - and check the box that says 'Automatically reply to messages when they are received.'
You can now customise the message and once complete, click on Done. Also see: How to downgrade iOS 10