While Feb-RU-ary is still considered the standard pronunciation, most dictionaries recognise the pronunciation of February without the first “r” (“Feb-U-ary”) as an acceptable variant.
In his Big Book of Beastly Mispronunciations (2005), Charles Harrington Elster defends the traditional pronunciation. Yet, in common speech, the shortest month has long been abused.
The loss of the first “r” in the pronunciation of February is (in part) the result of a process called disimillation (or haplology), where one of two similar sounds in a word is sometimes changed or dropped to avoid the repetition of that sound. (A similar process sometimes occurs with the pronunciation of library.)
The standard pronunciation of February takes considerable effort, and in normal rapid speech we are likely to drop the first ‘r.’ Also, the pronunciation of January has probably contributed to the simplified pronunciation of February.
There are, of course, many discrepancies between spelling and pronunciation in English. As David Crystal reminds us in The English Language:
Speech came first, in the history of our species, and English spelling hasn’t been a good guide to pronunciation for hundreds of years.