I am using Outlook on an exchange server and when I had my i Phone 5s, mail would happily update all the time i.e.
For example, 'apps gone free' would say 'your new apps gone free are ready' at about 6 PM every evening and my BBC news app would put messages on the screen and make the BBC news sound when a new piece of news came out, that no longer happens on my i Phone 6, although I believe I still have the same settings as on the i Phone 5s. I believe that is where you will find the "fetch new mail" option, although it may say something else.
Sorry I can't be of any more help to you, but I've never encountered that particular problem.
If it is an exchange email that you are having a problem with, I might suggest deleting that account from your phone and then setting it up as a new account again if that is possible.
If you are still having this issue, you can try updating your mail server in your iphone.
For example, you can use several different values for the email server, such as the server's hostname or the server's IP address.
Even though the following article is about FTP, it describes several different values you can use as a server name. If you have remove and re-added the email account and it is still not working, I recommend contacting our support department via phone, chat, or email to ensure that the settings that you are entering are correct.The same principle applies for both ftp and email (unless you are connecting securley using secure##.inmotionhosting.com). I am connecting securely so the solution you provide do not apply to me. If the settings are correct and you have removed/re-added the account to your phone, I recommend contacting Apple for further assistance as something within your phone may be corrupted. What worked for me (just a few minutes ago) was to go to Setting, then General, and at the bottom tap Reset, then on next page I hit reset network settings. The phone shut off for about five minutes and when it came up all my emails were fine.If you are still having the, "mail server is not responding" problem, please try a different I am having the same issue, my iphone mail client stoped working since two weeks ago. I was desperate, but you may want to first find out what hitting this reset does. Hello all, I have another problem with the upgrade to my new i Phone 6.Everything works fine now and IOS 8.1 looks ok to me.However, I am having a problem getting mail to upload when the 'mail' icon is closed.