In the United States, the most common pronunciation is feb-yoo-air-ee. Both Merriam-Webster and American Heritage dictionaries consider the common pronunciation correct, along with the less common, more traditional standard feb-roo-air-ee.
This gets fans of the traditional standard all worked up. But the loss of the first r in February is not some recent habit propagated by lazy teenagers. People have been avoiding that r for at least the last 150 years, and probably longer than that. Given certain conditions having to do with word stress and the other sounds in a word, we simply do not like to have two r's so close to each other. The name for the linguistic process where one sound drops out because another of the same sound is too close to it is dissimilation, and it affects lots of languages.
Consider your pronunciation of the following words, and be honest about whether you really say the r's in parentheses: su(r)prise, gove(r)nor, pa(r)ticular, be(r)serk, paraphe(r)nalia, cate(r)pillar, southe(r)ner, entrep(r)eneur, p(r)erogative, interp(r)etation. Not everybody drops these r's, but at the same time, nobody seems to get too upset when they hear others do it.
There are, however, a few cases of r dissimilation that get people very worked up, namely, lib(r)ary and Feb(r)uary. Lib(r)ary attracts attention due to its association with commonly disparaged dialects. Feb(r)uary only seems to attract attention when someone asks what the proper pronunciation should be.
Here's a little help from the (parody!) guide Pronunciation Manual:
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