I've been faffing round with getting outgoing emails to send on an "other" IMAP account for the past week since I upgraded to iOS 7.1.1
I have just managed to get things working on my iPhone 4S by doing the following;
Settings >> Mail, Contacts, Calendars >> select email account >> Account >>
Under "OUTGOING MAIL SERVER" >> SMTP >> Primary Server >>
Delete the Password and under Authentication, select something other than Password. Back out, and tap Done to verify. It won't, and after a few minutes will give an error message.
Go back in to Authentication and set it to Password again. Back out.
Re-enter the Password (also for me, SSL is on and Server Port is 587) and tap Done to verify.
When verification is complete, work your way back out of the settings tapping Done in the top right corner whenever its available.
Press the home and off/one button together to shut down the iPhone.
When it starts backup, outgoing SMTP mails send as normal (for me anyway).
Phew!
I'm heading home now to try this again on my iPad mini retina, and also on my sister's iPhone 4S, so fingers crossed that it will work there too. I'll confirm later what happens.