I heard someone pronounce UI as yooey. I guess this comes from the pronunciation of GUI, which is gooey. How common is this compared to yoo-eye and user-interface?
I've heard "user interface" used quite a lot, but never "yooey" - I'd say the latter was nonstandard.
Personally I'd pronounce it "yoo eye" in technical circles, and "user interface" in less technical circles.
UI is pronounced as "you I"
I work in IT industry and have never came across a different pronunciation.
Youey is informal, and is acceptable for relaxed chat or friends. But for a technical audience you should probably stick to expanding to User Interface for the first use and 'You Eye' afterwards.
This helps people who are not technical (who may not have come across the term before) and also people who don't speak English as a first language, where the initials U and I may have very different pronunciation. ('Uh ee'?)
 I do this all the time, and my friends find it endearing, or annoying, I'm never sure which. I also say 'wee-fee" for WIFI.  Sorry for not using ASCII IPA.
OK, I know this is old, but I have to interfere. I could let my resume speak to the kinds of circles I've been in, but even that's not enough. I've worked with everything and everyone in every field from basic coder monkey to building boards and writing drivers for them. From installing custom dvr's to law enforcement vehicles of every level, to setting up sound boards for an arena of 20+K, I've had a hand in everything. Point being, UI pronounced yoo-ee, has been pretty common for a few years now.
Now, MAYBE back in 2012, for quite a many of you, especially in engineering only fields, yoo-eye was considered more formal. However, as my experience has seen, as far back as about 08-09, yoo-ee has been more used, do to the fact it sounds more "friendly" to the people in management and ownership positions. Often times, even when trying to be professional, I, as did many techs back in the day, learned, that while "yoo-eye" may "sound more formal", it often kept a barrier between us and our ignorant of electronics customer (aka, your boss, usually). Thus, saying "yoo-ee" became more preferable as it keeps it sounding a little less informal, thus making it more formal in use by way of relationship to the customer. In other words, unless you're just in an engineering circle, which is usually small and made up of know-it-alls who really know a lot about one thing, (j/k ... maybe) "yoo-ee" is not only common, but often preferred, even in presentation to high scale customers.