The short answer is that 'Febryuæri' involves a consonant cluster that is not allowed in modern English.
In Old English, the name of this month was not February at all, but the aptly-titled 'Solmonað', 'mud-month'. Sometime after the Norman Conquest, the Old French 'Feverier' was borrowed into English, which in turn became 'Feoverel'. Thus the earliest names of this month in English did not have a 'bru' sequence at all.
During the 15th century, many words inherited from Middle English started to be given classicising spellings: 'oliphaunt', for example, became elephant (but was still pronounced 'olifont'), and likewise Feoverel became February in imitation of Latin 'Februarius'. Because many such classically-spelled words were not in particularly wide use, literate speakers often knew them solely from written texts and thus they often became pronounced with spelling-pronunciations.
What is unusual about all modern pronunciations of February, then, is that they came into use not by the normal means of one speaker uttering the word to other speakers, but via the historically exceptional and linguistically secondary method of writing. At the same time, the English language was undergoing the Great Vowel Shift, one of whose features entailed that the long vowel 'u:' was split into a glide 'y' and the vowel 'u'. With February, this had the effect of creating a cluster otherwise completely unattested in English: Middle English 'febru:æri' became Modern English 'febryuæri'. Different speakers resolved this problem in different ways: some deleted the 'r' to produce 'febyuæri', while other speakers – probably the minority – deleted the 'y', to produce 'februæri'.
Thomas Wier, linguist and Caucasologist
It's called regressive assimilation. There's stuff going on after the 'r' that ends up changing it. The 'u' sound before the 'a' is a little awkward for English. The 'a' sound (æ) is completely on the other end of the throat from 'u', so when we're reaching for it, we end up putting a big 'w' in there to help make the transition. So really people are saying 'Februwary'.
Well, now, trying to get from 'brew' to 'where' presents a problem. For some people that big 'w' is enough and they keep the 'r'. Others chop that 'u' on the way down to a 'schwa' (ə), the laziest vowel. That makes 'Febrəwary'. Another is to make the 'r' sound more like the 'w' and to reduce the amount of articulatory effort. Sparing you the phonetic and phonotactic details, that amounts to a 'y' sound in English, yielding 'Febyuary', which may not have a 'w' in it as a result of getting all that mess out of the way.
Michael Billips, linguist and writer
This is an edited answer from Why do so many people pronounce February 'Feb-yoo-ary' instead of 'Feb-roo-ary'? which originally appeared on Quora: The best answer to any question. Ask a question, get a great answer. Learn from experts and get insider knowledge. You can follow Quora on Twitter, Facebook, and Google+.