We all still wear tunics today. But now we call them "T-shirts" and they're a lot shorter.
That's all a tunic has ever really been: a t-shirt of varying lengths, varying styles of cut, made of various kinds of fabrics (depending on what was available and appropriate for one's status), and having various kinds of ornamentation.
Working-class people tended to wear plain, serviceable tunics that came to approximately knee-length (for men) and the ankles (for women). Depending on status, other clothing may have been worn over the tunics (ie. Roman patricians wore togas over their tunics when they were in public or indoor formal situations). Some aristocratic men and all women wore ankle-length (or floor-length) tunics.
My own thoughts as to why we (in the collective Western society sense) shifted from tunics to shirts come down to two factors:
1. Technology. Better tools to make and work threads, dyes, and fabrics would lead to more complex types of clothing. It's not hard to make a basic T-shirt (just ask any SCA member who has had to make a T-tunic). But a modern shirt? I for one wouldn't care to try.
2. Wealth. What major event happened around the time that middle and upper European classes shifted from tunics to shirts?
The first documented post-Viking voyages to the New World, in the late 1400s. As the Spanish, English, French, Portuguese, Dutch, and smaller European powers explored and acquired wealth from exploiting the resources of the New World, they wanted/needed new ways to show off their wealth.
Now how could a shirt be considered a form of wealth? By the embroidery, for one. Even today, some kinds of embroidery threads are quite expensive, and having a shirt liberally decorated with such embroidery and designs (men did the designing back then, btw) was a way of saying, "See what I can afford!" Add gemstones to the equation (especially for royal ladies and courtiers of both sexes), and you get a really complicated-to-make, expensively-decorated piece of clothing to wear at court and parties and show off your personal and family riches. Putting these ornamentations on a tunic doesn't carry the same sense of "wow!"
Also to do with embroidery and sewing gems onto clothing came lace. Lace on a tunic looks ridiculous. But lace on a shirt can look spiffy, if used in moderation (in a modern sense, at least). But the courtiers and wannabes during the Age of Exploration wanted to impress people, so they went in for ever more extravagant displays of lace, to the point where it was difficult to move their heads (and it must have itched abominably).
So to sum up, people generally sacrificed comfort and practicality in order to show off their wealth and impress people.
And we're still doing it (as I read in the OT thread about business suits). BTW, this is applicable to both men and women. I've had to wear blouses and suits and dresses (all modern descendents of tunics) to impress prospective employers - and hated every second of it.
Personally, I'd like to see the tunic - the original kind of tunic - make a comeback. They're so comfortable...