Standard Unicode Disclaimer: This writeup uses a bit of Unicode. If the Greek letters appear on your screen as boxes or question marks, you may need to tweak your browser.
Though it is impossible to know for certain how any language was pronounced thousands of years ago, it is clear that the pronunciation of ancient Greek underwent dramatic changes between the time of Plato and the rise of the Roman Empire four centuries later (just as English pronunciation changed significantly between the time of Shakespeare and today). Of course, people in different regions spoke different dialects of Greek in all time periods; just as English varies wildly between countries and even within cities, so also Greek was a rich and vibrant language at all points in its history. Still, there was an educated Attic dialect of Greek spoken by aristocrats in Athens which contemporary historians tend to treat as something like the "RP of the ancient world."
The dialect that was used by the authors of the New Testament is significantly different from this flavour of classical Attic Greek. The New Testament dialect is called koine, and, so far as modern historians can gather, many of the sounds that were easily distinguished in the classical period started to become confused with one another in koine. In koine Greek, the vowels η (eta), ι (iota), and υ (upsilon), along with the diphthongs ει, οι, υι, and ῃ, all began to slide toward an "ee" sound. (A similar phenomenon is found in modern Greek too, for reasons that are explained in another node.) The move from varied vowels to "ee" in Greek is called itacism.
The Effects of Itacism
Since scribes in the classical world often took dictation, the fact that certain words had recently become homophones meant that mistakes would often sneak into their written works. Some of these errors were like the "typos" that are familiar to us today -- who among us hasn't encountered a thousand writers who can't tell the difference between there and their? But some of the errors are more interesting (and occasionally more difficult to interpret) than that, since the homophones actually change the meaning of the sentence.
For example, in 1 Corinthians 15:54, most early New Testament manuscripts read "Death is swallowed up in victory" -- the word for "victory" here is νῖκος. However, in the manuscripts p46 and B, the sentence reads "Death is swallowed up in conflict" (νεῖκος). Similarly, in Revelation 4:3, John has a vision of a rainbow (ἶρις), but a few scribes render the word, weirdly, as "priests" (ἱερεῖς).
Some of the most serious problems with itacism in New Testament manuscripts have to do with pronouns. Since υ and ι were pronounced so similarly, the first-person plural ἡμεῖς ("we") closely resembled the second-person plural ὑμεῖς ("y'all") when spoken aloud. This may seem odd to an English speaker, since for us u and i do not seem especially alike. However, anyone who is familiar with the French u (found in words like sūr or attendu) will know that it's possible to have a sound in between.
Anyway, there are places in the New Testament where it's genuinely hard to tell from the manuscript evidence whether the author intended to use the first person or the second person. In Galatians 4:28, does Paul imply that we all are children of promise, or is the pronoun directed more pointedly to you, his readers? Does the author of the first letter of John include himself with his readers in his opening wish ("that our joy may be complete") or does he simply hope that their joy might be complete? The surviving manuscripts are so inconsistent that it is genuinely difficult to tell which reading was the original. If you have a good quality study Bible, look these passages up yourself: the footnotes will show you where there are textual uncertainties.
Though there are lots of causes of scribal error, including everything from incompetence and fatigue to poor working conditions, itacism provides a unique twist to the study of sources in the koine language.
Further Reading:
I drew most of these examples from the classic study by Bruce Metzger and Bart Ehrman entitled The Text of the New Testament, Oxford University Press 2005, which is well worth picking up if you find this stuff interesting.